Rabbi Moses Isserles (1525-1572,) in his Notes to the Shulchan Aruch of Joseph Caro, writes that for the sake of retaining good will and avoiding enmity Jews who find themselves in the midst of a Gentile holiday celebration should join the festivities.  He suggests that Jews should not seek out such celebrations, but he considers it proper for Jews to send gifts to non-Jews on non-Jewish holidays. 

This long standing ruling is of particular significance in modern societies where Jews are invited to attend Christmas parties and celebrations and are expected to exchange gifts with non-Jewish friends in celebration of the holiday.  Apparently the problem is an old one. 

What applies to Christmas applies equally to other holidays that are basically of a Christian religious nature: Halloween (All Saints' Day,) Valentine's Day, and the like.

Thanksgiving, despite its religious overtones, is not a Christian holiday and is celebrated fully by American Jews.