That's a very good question!
Certainly Orthodox Jews (especially the Charedim) do not observe Thanksgiving Day, as G-d has given them holidays of their own.
For Noachides Thanksgiving is a more complex issue. It is problematic because it is a religious holiday, and non-Jews are forbidden to invent new religious holidays, even if those holidays are based on the Noachide Laws themselves. However, we know from the TaNa"KH that non-Jewish kings have in the past proclaimed days of fasting in repentance for their sins. I suppose a day of thanksgiving would fall in the same category, though whether this should be a regular annual day or a reaction to the preaching of a Prophet I do not know.
This is the best I can tell you right now.