I have many non Jewish facebook friends who live in American that I keep in touch with Many wish me a Happy Chanukah and I feel an obligation to wish them a merry christmas as the 2 holidays occur at approximately the same time in year. Is this permitted according to halacha?

Shalom, Thank you for your question. The issue of sending greetings to a non-Jew on their religious holidays is mentioned in the Shulchan Aruch (Yoreh Deah, 148,9), where he writes that "it is forbidden to go into an idol worshipper's house on his festival day to great him with a "Shalom", but outside his house it is permitted. But one should [none the less] not give an over-enthusiastic greeting". The commentators there (see the Shach) discuss whether this law is limited to using the word "Shalom", or all greetings. The ruling is that one may give greetings which do not include the word "Shalom" as it is one of the names of G-d, but rather use wording such as "Good Day", "Long Life!", "Good luck in everything" etc. (see Zedah La'Derech, Machon Zomet, p. 197). So, it would be best for you to send "Holiday Greetings", or "All my love over the holidays!", rather than either using a greeting with G-d's name in it. Also, "holiday greetings" are better as it could mean the holiday of Chanukah. May you be blessed with a great and holy Chanukah, and "all the best to all your facebook friends over the holidays!" Blessings.