Are there things Jews are not allowed to do, per halacha, simply on the grounds that some non-Jews do it, and not because these things are bad in themselves?
(These non-Jews may have a bad overall reputation, may do other things we violently disapprove of, we may not want to be confused with them, etc.)
Yes. As discussed in Sahedrin 74b and codified in Halacha, there are "things Jews are not allowed to do, per halacha, simply on the grounds that some non-Jews do it, and not because these things are bad in themselves".
The Kitzur Shulchan Aruch in 3:2 brings some examples and guidelines:
אֵין הוֹלְכִין בְּחֻקּוֹת הַגּוֹיִים, וְלֹא מִתְדַמִּים לָהֶם, לֹא בְמַלְבּוּשׁ וְלֹא בְשֵׂעָר וְכַיּוֹצֵא בָהֶם, שֶׁנֶּאֱמַר, וְלֹא תֵלְכוּ בְּחֻקּוֹת הַגּוֹי. וְנֶאֱמַר, בְּחֻקּוֹתֵיהֶם לֹא תֵלֵכוּ. וְנֶאֱמַר, הִשָּׁמֵר לְךָ פֶּן תִּנָּקֵשׁ אַחֲרֵיהֶם.
We are not permitted to follow the ways of the gentiles, nor adopt their styles in dress or in hairstyle or similar things, as it is said: "You shall not follow the ways of the gentile." It is [also] said: "In their ways you shall not follow" It is [also] said: "Guard yourself lest you be ensnared to follow them."
לֹא יִלְבַּשׁ מַלְבּוּשׁ הַמְּיֻחָד לָהֶם לְשׁוּם גֵּאוּת וְהוּא מַלְבּוּשׁ שָׂרִים, וּלְדֻגְמָא הָא דְּאָמְרֵינָן בַגְּמָרָא, דְּאָסוּר לְיִשְֹרָאֵל לְהִתְדַּמּוֹת לָהֶם אֲפִלּוּ בִרְצוּעוֹת הַמִּנְעָל. שֶׁאִם הָיָה דַרְכָּם לִקְשֹׁר כָּךְ וְדֶרֶךְ יִשְֹרָאֵל בְּעִנְיָן אַחֵר אוֹ שֶׁהָיָה מִנְהָגָם לִהְיוֹת לָהֶם רְצוּעוֹת אֲדֻמּוֹת וְשֶׁל יִשְֹרָאֵל שְׁחוֹרוֹת מִשּׁוּם דְּצֶבַע שָׁחוֹר מוֹרֶה עַל עֲנָוָה וְשִׁפְלוּת וּצְנִיעוּת, אָסוּר לְיִשְֹרָאֵל לְשַׁנּוֹת.
You should not wear a garment which is specifically worn by them as a symbol of ostentation, such as a garment of high-ranking officers. For example, the Talmud states: that it is forbidden for a Jew to be similar to them even in regard to shoelaces; if their practice was to tie one way and the practice of Jews to tie another way, or if their practice was to wear red shoelaces and Jews wear black shoelaces because the color black indicates humility, submissiveness and modesty. [In all such instances] it is forbidden for a Jew to deviate.
וּמִזֶּה יִלְמַד כָּל אָדָם לְפִי מְקוֹמוֹ וְשַׁעְתּוֹ, שֶׁהַמַּלְבּוּשׁ הֶעָשׂוּי לְיוּהֲרָא וּפְרִיצוּת לֹא יַעֲשֶׂה הַיִשְֹרָאֵל כֵּן, אֶלָּא מַלְבּוּשָׁיו יִהְיוּ עֲשׂוּיִים בְּעִנְיָן הַמּוֹרֶה עַל הַכְנָעָה וּצְנִיעוּת.
From these examples everyone should learn how to apply these standards to his time and place. A garment designed for showiness or immodesty must not be worn by a Jew, but rather his clothing should be made in a style which suggests humility and modesty.
הָכֵי אָמְרִינָן בְּסִפְרֵי, שֶׁלֹּא תֹאמַר הוֹאִיל וְהֵן יוֹצְאִין בְּאַרְגָּמָן אַף אֲנִי אֵצֵא בְּאַרְגָּמָן, הוֹאִיל וְהֵן יוֹצְאִין בְּקוֹלְסִין (פֵּרוּשׁ, כְּלֵי זַיִן) אַף אֲנִי אֵצֵא בְּקוֹלְסִין, שֶׁדְּבָרִים הַלָּלוּ דִבְרֵי שַׁחַץ וְגַאֲוָה הֵם, וְלֹא כְאֵלֶּה חֵלֶק יַעֲקֹב, אֶלָּא דַּרְכָּם לִהְיוֹת צְנוּעִים וַעֲנָוִים, וְלֹא לִפְנוֹת אֶל רְהָבִים.
The following is mentioned in Sifrei: You should not say that since they go out with scarlet I shall go out with scarlet, since they go out with kulsin (the word kulsin meaning weaponry) I also shall go out with kulsin, because these practices are indicative of arrogance and haughtiness which are not the heritage of Jacob. Rather, our heritage demands of us to be modest and humble, and not be influenced by the haughty.
וְכֵן כָּל דָּבָר שֶׁנָּהֲגוּ לְמִנְהָג וּלְחֹק דְאִכָּא לְמֵיחָשׁ שֶׁיֵּשׁ בּוֹ שֶׁמֶץ עֲבוֹדָה זָרָה, לֹא יַעֲשֶׂה הַיִשְֹרָאֵל כֵּן.
Similarly, any custom or statute of which there is a suspicion of idolatrous intent or background should be avoided by Jews.
וְכֵן לֹא יְגַלַּח וְלֹא יְגַדֵּל שַעֲרוֹת רֹאשׁוֹ כְּמוֹהֶם, אֶלּא יְהֵא מֻבְדָּל מֵהֶם בְּמַלְבּוּשָׁיו וּבְדִבּוּרוֹ וּבִשְׁאָר מַעֲשָׂיו כְּמוֹ שֶׁהוּא מֻבְדָּל מֵהֶם בְּמַדָּעוֹ וּבְדֵעוֹתָיו. וְכֵן הוּא אוֹמֵר, וָאַבְדִּיל אֶתְכֶם מִן הָעַמִּים.
Similarly, you should not cut your hair or style your hair as they do, but rather you should be distinct, in your clothing and speech and all other endeavors just as you are distinct in your perspectives and concepts. Similarly, it is said: "I have set you apart from the nations."
Just a couple of clarifications:
Not "some non-Jews do" but "what's commonly accepted in a specific place". If a hundred people in NY do or wear something special we should not be worried.
This prohibition is a spin-off from the prohibition of idolatry and adultery (to distance ourselves from the possibility of marrying Gentiles). This is not a "stand-alone" prohibition, "just G-d's decree.
Seemingly this Halachah only applies to "bad" practices, but if they do something to enforce the 7 laws that aligns with our Torah, for example, obligate wearing a Kippah, or extra-long skirts we don't do the opposite just to be different.
Or if the government representatives wear suits that doesn't mean we shouldn't. Or playing baseball, even professional for one's health and living.
This Halachah has a flip side (as many other conducts), where a person that openly exaggerates in being different, and wears a robe, that is called "מחזה כיהורא" - looks like vanity, like being better in things that most if not all Rabbis observed.
I never saw a Rabbi with Jewish shoelaces or demonstrational Jewish shirt or socks.
This Halachah is bound to more "severe" Halochos, like self-preserving or making living etc, and in cases of contradiction the latter win (as the Gemmorah rules with יושב במלכות).
