I've seen the answers here that say a non-Jew may not keep Shabbat, although I would definitely take issue with the green check-marked answer there that uses Sefer Shofetim, Hilkhot Melakhim uMilchamoteihem 10:11-12 for this assertion--the cited text does not even imply such a thing, instead it discusses the idea that a בן-נח who wishes to perform a mitzva for the reward of doing so, should not be prevented. In Pereq 9, Ramba"m writes that an idolater (עכו"ם) should be scourged if he attempts to keep Shabbat or studies Torah, or creates any religious practice, telling him he is liable for death for doing so -- but it doesn't say anything about requiring a גר תושב, nor anyone else, to deliberately violate Shabbat ח"ו.
From my understanding, Ramba"m is concerned that the aku"m should know that he is liable to punishment from Heaven if he keeps mitzvot, because by doing so he is indicating a desire to be attached to the covenant...his scourging is to prevent him from damaging himself, not to prevent him from improving himself, as Ramba"m expounds in pereq 9. No such pronouncement is made against a ger toshab.
Where does the teaching come from that a non-Jew (a בן נח) should be compelled to violate Shabbat, even in a small way? Certainly a non-Jew should not say the full berakha for qidush, but a non-Jew not keeping Shabbat completely as a Jew does, is a very different thing from requiring the non-Jew to actively violate Shabbat.
Edit: To be very clear here, I see it as a grave error to assert that the non-idolator non-Jew should be compelled to violate Shabbat -- Ramba"m in Hilkhot Melakhim uMilchamoteihem and Resh Laqish in the gemara of Masekhet Sanhedrin 58b (erroneously referred to as מ"ש by me in comments below--I was trying to shorten stuff up and was thinking Shabbat instead of Sanhedrin) are almost certainly discussing idolaters who are known to not have Jewish ancestry. The notion of compelling someone who has gone off the derekh, or whose great-great-great grandparents did, to violate Shabbat, is horrifying. It is the same greatest problem with the notion of the "shabbos goy"--you don't know who is and isn't halakhically Jewish. Just because they don't know they're Jewish doesn't mean you know that they're not Jewish. By promulgating the teaching that a person who is not known to be a Jew should violate Shabbat, you could very well be compelling an unwitting halakhically Jewish Jew to violate a precept of Torah observance mi-Sinai. What possible benefit is there in doing so for anyone?
