I'm trying to draw an analogy between the written and the oral Torah. After the Mishna was signed, the Sages continued to use extra-Mishnaic sources forever (Toseftah, Midrashim, etc).
After the 24 books of the Tanach were finalized/canonized, does the Talmud (the Sages) include citations or references to extra-Tanachic books (Ben Sirah, 1-2- Maccabees, Sefer Zerubavel, Sefer Chanoch, Sefer Raziel, and many others, thanks Yaacov Deane)? Or none of them are mentioned, but the 24 books?
As @TamirEvan stated in the comments, Ben-Sira appears a few times in the Talmud, for example:
"The verse states: “All the days of the poor are terrible” (Proverbs 15:15). The book of ben Sira says: Also the nights are terrible, as then the poor person worries. The poor person’s roof is among the lowest roofs in the city, and in the elevation of the hills is his vineyard, at the highest point, as those are of the lowest quality and consequently the least expensive places for each. From the rain on the roofs of the entire city, water will flow down to his roof and dampen it, and the soil of his vineyard is eroded by the rain and swept down to other vineyards."
and Ketubot 110b:
"The mishna taught: Rabban Shimon ben Gamliel says that a pleasant residence tests the individual. The Gemara asks: What is the meaning of the term tests in this context? The Gemara explains: This is in accordance with the opinion of Shmuel, as Shmuel said: A change in one’s eating habits [veset] or in one’s place of residence is the start of intestinal disease. Similarly, it is written in Sefer Ben Sira: All the days of the poor are terrible. And yet there are Shabbatot and Festivals, when even the poor eat well. Once again, the Gemara answers: This is in accordance with the opinion of Shmuel, as Shmuel said: A change in one’s eating habits or in one’s place of residence is the start of intestinal disease, and as a result the poor suffer even from a change for the better. Since the Gemara quoted from Sefer Ben Sira, it cites the rest of the passage concerning the terrible days of the poor. Ben Sira says: Even the nights of the poor are bad. His roof is at the low point of the roofs, i.e., his residence is at the lowest point in the city, and his vineyard is at the mountain peaks, at the highest point of the slope, which means that the rain of roofs washes down to his roof, and the soil of his vineyard to other vineyards, i.e., the rain washes away the soil in his vineyard and carries it away to the vineyards below."
In both places, it seems that the gemara used Ben-Sira as an "asmachta" (אסמכתא) much in the same way as it usually uses psukim from the Tanach. However, in other places, it says one should not read Ben-Sira.
Rabbi Dr. M. Z. Segal in his edition of Ben Sira, wrote that even teachings such as "Rabbi Levitas a man of Yavneh said: be exceeding humble spirit, for the end of man is the worm." (Avot 4:4) are paraphrases of teachings of Ben Sira, who wrote long before "Abase pride very exceedingly; For the hope of man is the worm" (Ben Sira 7:16) (Compare the Hebrew).
The Talmud and midrash further mention two books that we don't currently have but were included in the apocrypha (Sefarim Chitzonim - External Books, i.e. books that should not be studied) in the time of the gemara: Ben La'anah (Yerushalmi Sanhedrin 50a) and Ben Tigla (Kohelet Rabbah 12:12).
Furthermore, Rabbi Prof. Shnayer Leiman in one of his classes on the Dead Sea Scrolls, mentions that Rashi quoted a verse that doesn't appear in Tanach: "אם תעזבני יום יומים אעזבך - If thou forgettest Me one day, I will forget thee two days". Rashi says it's from a certain "megillah" (scroll). Both the midrash in Sifrei Devarim 48:8 and the Yerushalmi in Brachot 68a say it's from "Megillat Chassidim". It's possible that this scroll was also an apocryphal work in the time of the gemara.
In an essay titled "The Inverted Nuns", Rabbi Leiman quotes Midrash Mishlei 26:24 (emphasis mine):
"The verses beginning: "When the ark was to set out" (Num 10:35) are marked above and below. Rabbi [Judah the Prince] said: It was a separate book and withdrawn."
And then he presents the (reluctant) opinion of Rabbi Shaul Lieberman that this midrash may be referring to a book known as "Sefer Eldad U'Meidad", because there's another midrash that was found in the Cairo Genizah and first published in the late 19th century by Elkan Adler which says (emphasis mine):
"Some Midrashim expound this differently. They state: Why did the sages place inverted nuns before the verse: The people took to complaining (Num 11:1)? The sages thereby declared: The entire Torah consists of the prophecy of Moses except for these two verses (i.e., Num 10:35-36) which are from the prophecy of Eldad and Medad. Therefore they were enclosed with a curved nun and inserted into the Torah." (original Hebrew at the bottom of pg. 705)
Rabbi Leiman notes in his essay that a "Book of Eldad and Meidad" appears in lists of apocryphal or pseudepigraphal works but is currently non-extant, although per these sources, it seems that this work may have been incomparable to other External Books, because it was actually held, at least by some authorities, to have been written by Eldad and Meidad (on what this book may have contained, see Sanhedrin 17a and Sifrei Bamidbar 95) (it should be noted that Rabbi Leiman himself believes that both this formerly-lost midrash and another midrash call Midrash Chaserot V'yeterot both misinterpreted Midrash Mishlei, which he believes does not refer to a lost book authored by Eldad and Meidad) (however, Prof. Baruch A. Levine wrote an essay attached to the end of the PDF of Rabbi Leiman's essay, in which he argues against Rabbi Leiman's conclusions).
In Jahrbuch IX, pg. 13-15, Rabbi Dr. Chanoch (Heinrich) Ehrentreu wrote that he thinks the verse from Yevamot 86b, brought by @moses in his answer, "ושוטרים הרבים בראשיכם", is from an apocryphal work, specifically, in his opinion, the original Hebrew version of Maccabees I, which he believes is the same work as the "Megillat Beit Chashmonai" mentioned in the Halachot Gedolot as having been written by sages from Beit Hillel and Beit Shammai. There are a number of other interesting interpretations of the verse. For example, Rabbi Amos Sofer in Sinai, vol. 98, thought it might be from one of the Dead Sea cults or perhaps the Essenes (which may or may not be the same group).
Other than this verse, there are multiple verses in Chazalic works that don't appear in Tanach. I've managed to find 23, but there are likely others. Most commentators think that they are either shortened versions of extant Tanachic verses or come from Ben Sira, but a few - mostly modern scholars - think that at least some come from apocryphal works.
There are also parallels that can be drawn between certain apocryphal works and midrashim. For example, Professor Yehoshua Grintz (in his edition of Sefer Yehudit), Chanoch Albeck (Beresheet Rabbti, pg. 15) and Adolf Neubauer (Book of Tobit, pg. 38) all write that the midrash in Tanchuma Ha'azinu 8 is clearly the story of the Book of Tuviyah with the names and much of the plot removed, while Rabbi Dr. Moshe Gaster found an edition of Megillat Ta'anit which includes the story of Yehudit. The difference between this version of the story of Yehudit and other Chazalic mentions of the story is that this version adds a "Yom Tov" that they celebrated which is similar to the celebration mentioned at the end of the Aramaic version used by Jerome in the Vulgate (this "Yom Tov" isn't mentioned anywhere else) (see here for more information).
Adding to @Harel13's answer, four points:
אמר רב חסדא בתחלה לא היו מעמידים שוטרי' אלא מן הלוים שנאמר (דברי הימים ב יט, יא) ושוטרים הלוים לפניכם עכשיו אין מעמידין שוטרים אלא מישראל שנאמר ושוטרים הרבים בראשיכם:
Rav Ḥisda said: Initially they would establish officers over the people only from among the Levites, as it states: “And the officers, the Levites, before you” (II Chronicles 19:11), but now they establish officers only from among the Israelites, as it is stated: And the officers of the many at your heads.
There is no such verse anywhere in Tanach (Tos' Yeshanim S.V. שנאמר), Ben Sira or any Apocrypha. (See Yad Malachi Klalei Hatalmud #283)
א"ל רבא לרבה בר מרי מנא הא מילתא דאמרי אינשי מטייל ואזיל דיקלא בישא גבי קינא דשרכי אמר ליה דבר זה כתוב בתורה שנוי בנביאים ומשולש בכתובים ותנן במתניתין ותנינא בברייתא כתוב בתורה דכתיב (בראשית כח, ט) וילך עשו אל ישמעאל שנוי בנביאים דכתיב (שופטים יא, ג) ויתלקטו אל יפתח אנשים רקים ויהיו עמו ומשולש בכתובים דכתיב (בן סירא יג, יד) כל עוף למינו ישכון ובני אדם לדומה לו תנן במתני' כל המחובר לטמא טמא כל המחובר לטהור טהור ותנינא בברייתא רבי אליעזר אומר לא לחנם הלך זרזיר אצל עורב אלא מפני שהוא מינו
Rava said to Rabba bar Mari: From where is this matter derived whereby people say: A bad palm tree strolls and goes to be among a grove of barren trees, i.e., bad people seek out other bad people? Rabba bar Mari said to him: This matter is written in the Torah, repeated in the Prophets, and triplicated in the Writings, and we learned it in a mishna, and we learned it in a baraita. Rabba bar Mari explains each of the sources. It is written in the Torah, as it is written: “And so Esau went to Ishmael” (Genesis 28:9). It is repeated in the Prophets, as it is written: “And there were gathered vain fellows to Yiftah, and they went out with him” (Judges 11:3). And it is triplicated in the Writings, as it is written: All fowl will live with its kind, and men with those like him (Book of Ben Sira 13:17). We learned it in a mishna (Kelim 12:2): All that is attached to that which is ritually impure is ritually impure; all that is attached to that which is ritually pure is ritually pure. And we learned it in a baraita: Rabbi Eliezer says: Not for naught did the starling go to the raven but because it is its kind, as it too is a non-kosher bird.
Here we see not only that the Gemara quotes ben Sira, but that it treats it as "משולש בכתובים"! Furthermore, the source from ben Sira can be found in any ש׳׳ס in the מסורת הש׳׳ס.
וְאָמַר רַבִּי יוֹחָנָן, מַאי דִּכְתִיב: ״וַיְהִי בִּימֵי שְׁפוֹט הַשּׁוֹפְטִים״? דּוֹר שֶׁשּׁוֹפֵט אֶת שׁוֹפְטָיו; אוֹמֵר לוֹ: ״טוֹל קֵיסָם מִבֵּין עֵינֶיךָ״ – אוֹמֵר לוֹ: ״טוֹל קוֹרָה מִבֵּין עֵינֶיךָ״.
Rabbi Yoḥanan says: What is the meaning of that which is written: “And it happened in the days of the judging of the Judges” (Ruth 1:1)? This indicates a generation that judged its judges. If a judge would say to the defendant standing before him: Remove the splinter from between your eyes, meaning rid yourself of some minor infraction, the defendant would say to him: Remove the beam from between your eyes, meaning you have committed far more severe sins.
This is a direct quote from the Book of Matthew, chap 7:
1 "Do not judge, or you too will be judged. 2 For in the same way you judge others, you will be judged, and with the measure you use, it will be measured to you. 3 "Why do you look at the speck of sawdust in your brother's eye and pay no attention to the plank in your own eye? 4 How can you say to your brother, `Let me take the speck out of your eye,' when all the time there is a plank in your own eye? 5 You hypocrite, first take the plank out of your own eye, and then you will see clearly to remove the speck from your brother's eye.
EDIT:
R. Yochanan is actually coming off of an even earlier source in Chazal, in Arachin 16b:
תניא, א"ר טרפון: תמה אני אם יש בדור הזה שמקבל תוכחה, אם אמר לו טול קיסם מבין עיניך, אמר לו טול קורה מבין עיניך.
It is taught in a baraita that Rabbi Tarfon says: I would be surprised if there is anyone in this generation who can receive rebuke. Why? Because if the one rebuking says to him: Remove the splinter from between your eyes, i.e., rid yourself of a minor infraction, the other says to him: Remove the beam from between your eyes, i.e., you have committed far more severe sins.
Being that this last source an be traced as far back as R. Tarfon, who lived soon after the destruction of the second Beit Hamikdash, and was known for some extremely anti-christian Halachot (Shabbos 116a, R. Tarfon says that the Evangilium should be burned) it seems unlikely that he would be quoting from them. See also here, which discusses the history of the phrase, and concludes that a) the correct text should say שיניך, your teeth, rather than עיניך, your eyes, and b) that it was indeed a proverbial phrase.
