It's my understanding that Shabbat was given to the Jews alone. Past that, none that I can think of. Pesach is a bit controversial, but not sure if it falls into actual forbidden territory (Sophiee will know, I'm sure). And how could any sort of prohibition be enforced anyway?

OTOH, it occurs to me that Shabbat is probably the greatest gift we as a people have given to civilisation.

Gentiles are welcome to keep Shabbat even by frum standards so long as they turn on a light or do one thing that would be forbidden for Orthodox Jews on Shabbat--so they'd technically be breaking it, though they would for all intents be keeping it (unless you want to get really fiddly on the details), and that's fine. At least that's what I've heard from a number of frummies and Orthodox rabbis, and I've no reason to disbelieve them.

Then in Torah there's the whole thing about how even your servants, animals, and resident aliens (I think that's the meaning of 'gerim' in that passage, as Uri's noted, the word can mean a few things) are supposed to keep Shabbat too because it's a day of rest for everyone, but maybe that only applies in Israel? Rabbi Daniel? I'm sure I've missed some context on that one.

My guess would be that the few non-Jews who do keep Shabbat (I've never met one, actually) keep it along the same lines as those of us somewhere in the Reconstructionist/Renewal/Conservative camp do, and here there are various levels of observance from strict to just lighting candles and having a 'family night' on Friday nights and everything in between. The idea being that something is a whole lot better than nothing.

Outside the Orthodox community, it's not a big issue, at least not that I'm aware of. It seems like the more important thing would be to encourage Shabbat observance amongst Jews rather than discourage it amongst non-Jews.