Being a completely non-practicing member of the faith, I could be wrong here, and if so I'm sure someone will correct me, but to my way of thinking.....

The only holiday on which it would be appropriate to wish a Jewish person a "Happy" would be Rosh Hashanah, the Jewish New Year. But I don't think any American Jews would be even the slightest bit offended if their gentile friends or associates failed to give them a Jewish New Year's greeting.

With respect to Hanukkah, or Hanukah, or Chanukah (all three spellings are acceptable), this is a holiday which celebrates the victory of the Maccabees over the Syrians (who I think were a Jewish tribe who had been enslaved by the Syrians at the time) in 165 B.C., and the rededication of the Temple at Jerusalem (Don't ask me -- I got that straight from the dictionary. You now know pretty much as much about it as I do).

It is traditional for adults to give children presents, and since it usually falls somewhere around Christmas, it all works out rather neatly I think.

While an adult would wish a child a "Happy Chanukah" when giving him or her a gift, I have seldom heard adults wish each other a greeting, and again, I don't think a Jew would expect such a greeting from a gentile.

As far as Yom Kippur goes, that is the "Day of Atonement", on which practicing Jews fast all day to atone for their sins of the previous year.

Because of the holiday's solemnity, any type of holiday greeting in this case would be quite inappropriate.