"Although it is clear enough that Jewish men should not touch women for reasons of niddah"
This is not entirely exact. Sexual relationship is prohibited and punished by Karet. Touching is not prohibited by itself, but as a guard rail, distancing between spouses in prohibited period are rabbinic. A problem of touching Nidda exists regarding to Tum'a and Tahara but is not included in the scope of the question. But a married woman (Eshet Ish, prohibited because married), even if clean from a nidda point of view, is not allowed to touch, e.g. shaking hand, rabbinicly (at least) because her status of Eshet Ish.
See Sefer Hachinuch 188:
שלא להתעדן [נתעדן] באחת מכל העריות. והן קרובות ואשת איש ונדה, ואפילו בלא ביאה כגון חבוק ונשוק וכל הדומה לאלו הפעלות הרעות שמעמיקין בהן בעלי הזמה ההולכים אחרי ההבל ויהבלו, שנאמר (ויקרא יח ו) איש איש אל כל שאר בשרו לא תקרבו לגלות ערוה. ופרשו, כאלו אמר לא תעשה שום קריבה שהיא הגורמת והמביאה האדם לגלות ערוה. וכן דרשו זכרונם לברכה (ספרא אחרי יג טו וכא) לא תקרבו לגלות, אין לי אלא שלא יגלה מנין שלא יקרב? תלמוד לומר ואל אשה בנדת טמאתה לא תקרב, אין לי אלא נדה בבל [תקרב ובל] תגלה, מנין לכל העריות? תלמוד לומר לא תקרבו לגלות, ושם נאמר (שם כט) ונכרתו הנפשות העושות, שמא תאמר יהיו חיבים כרת על הקריבה לבד? תלמוד לומר העושות ולא הקרבות.
It's prohibited to have a "pleasant contact" with a prohibited one.
ונוהגת בכל מקום ובכל זמן בזכרים ונקבות,
This Mitsva apply equally for men and women.
halachic reason that a Jewish woman could not touch a non-Jewish man?
--> Yes, here is the source.
If we read the text in Sefer Hachinuch and poskim, accurately, shaking hands is a gesture of rapprochement. The Sefer Chassidim quoted by hazoriz is congruent whit Ramban ((On Rambam, Sefer Hamitsvot LT, 353; Ramban in his "objections" disagreed with the fact that there is a Scriptural prohibition, but agrees with this rule as rabbinic.).
halachic reason that a Jewish woman could not touch a non-Jewish man?
--> Prohibition to marry is scriptural, and regards women as men, so it is prohibited (lechaora)
I am referring to non-affectionate touch, such as sitting next to someone or passing
--> A non-affectionate touch is out of the spectrum of the prohibition above addressed.
###Sifra Debe Rav
The Chinuch above cited (almost an excerp from Rambam) quoted a Berayta from Torat Qohanim. The famous discussion which is being held between Rambam and Ramban is surrounding this Berayta:
פרק יג
[יח,ו]
(א) איש -
מה תלמוד לומר איש איש?
להביא את הגוים שיהיו מוזהרים על העריות כישראל.
לא תקרבו -
מה תלמוד לומר?
לפי שנאמר: איש איש שיכול אין לי אלא איש ממש שהוא מוזהר על ידי אשה, אשה מוזהרת על ידי איש מנין?
תלמוד לומר: לא תקרבו הרי כאן שנים.
[יח, יט]
(ב) ואל אשה בנידת טומאתה לא תקרב לגלות ערותה -
אין לי אלא שלא יגלה מנין שלא תקרב? תלמוד לומר לא תקרב אין לי אלא נידה בל תקרב בל תגלה. מנין לכל העריות בל תקרבו ובל תגלו. תלמוד לומר לא תקרבו לגלות. אני ה' אני נאמרן לשלם שכר.
"Don't go too near"(plural form) what is the meaning? A warning not only for men but also for women.
In the Torat Qohanim (Sifra) we see comparison between rules concerning Nidda relationship prohibition and rules concerning all Arayot. The sifra makes equivalence with each other.
Poskim
See Bet Yosef YD 195. He explains that the distancing must be greater with his wife Niddah because of a greater facilitation to engage relationship. The difference between nidda-wife and Eshet Ish concerning the touching problem is not qualitative, but quantitative. __Exemple given, to check pulse is Negia Bealma (a simple touching) with an other prohibited women and is allowed for a physician, but with his own wife Niddah, this can lead to greater rapprochement and is prohibited. So it is prohibited scripturaly according to Rambam (following the Bet Yosef understanding. Shach 157 sk 10 disagree with this explanation in Rambam).__
Rabenu Yona agrees with the Rambam opinion (following Bet Yosef) that to get closer to nidda is a case of Yehareg Veal Yaavor (let him be killed and do not infringe). Rabenu Yona's opinion is based on 2 premises which are specific to him. 1.- The rule of Yehareg Veal yaavor is for all women with which the intercourse is punishable of Kareth (as Ran in Chidushin Sanhedrin 74a), including Niddah. 2.- There is Yehareg Veal Yaavor with Kirva (to get closer) as we see in Gemara with Nathan Tsutsita (Sanhedrin 75a, Gemara explains that Chachamim do not allow to this man to look a women {2 opinions in Gemara if Eshet Ish (married) or celibate, anyways all opinions ruled this Gemara concerning Eshet Ish (אין מתרפאים באביזריהו דעריות)} even if he can die if he do not see him.
.
Most poskim did not follow R.Y. in neither premises: concerning Yehareg, no Yehareg for Niddah (because Kiddushin Tofsim, some can marry a Niddah woman), and not concerning this kind of kirva (Shach in 195, 20 and 157, 10 learn in Rambam a touching clearly sexually oriented, even if the woman is his wife).To get closer but not explicitly sexually oriented.
We already said that the Shach YD 157, sk 10, YD 195, 20 said that the kirva prohibited scripturaly for Rambam is only when sexually oriented.
Kinds of physical contact
There are 4 degrees of touching. 1.- without rapprochement. 2.- without intrinsec rapprochement but between husband and wife. 3.- with no sexually oriented rapprochement. 4. with sexually oriented rapprochement.
Shaking hand
Hand shaking is the 3rd degree . Following the Shach it is prohibited Rabbinicly for a prohibited woman, Rambam ruled scriptural prohibition. For Ramban rabbinic. For Rabenu Yona scriptural and may be Yehareg Veal Yaavor. The sefer Chassidim seems to think that it is rabbinic. A Jewish woman is prohibited to shake hand to a non-Jewish man.
Shaking hands with gloves, without gloves
See Yerushalmi Sota 3, 1:
>וכהן מניח את ידו תחתיה ומניפה. ואין הדבר כאור. מביא מפה. ואינו חוצץ. ומביא כהן זקן אפי' תימר ילד שאין יצר הרע מצוי לשעה.
A Cohen puts his hand under hers and swings it. Is this not improper (hideous, according to Soncino's translation in Chulin)? He brings a towel. Is it not a "barrier". He brings an old Cohen, maybe even a young because the Yetser is not frequent for a very short time.
A separation between skins make the contact not hideous, may be the brevity, or the old age of the man too.
See Sefer Chassidim Nussach Parma 1266 (תתתסו) The commentary (I learned the Yerushalmi above from it) says that Sefer Chassidim prohibit only when the shaking of hands has a symbolic meaning of contract, because it seems as a concluding contract about sexual relationship. So he does not prohibit, so shaking hand with glove to greet persons of different gender may be allowed, at least with glove, according this author (but it is very impolite to shake hand with glove in France!).
Finally there is shaking and shaking. may be that a very fast and formal shaking is not rapprochement. But almost sure that it is. There is at least a Rabbinic prohibition. Chachamim said (Chulin 44b) "but the Sages have said: 'Keep aloof from anything hideous or from whatever seems hideous'!"
