I can assure you, I know the correct way of spelling the aforementioned holiday: חנוכה (assuming non-ascii characters are permitted).

That aside, there is no technically correct way of spelling Chanukah because Hanukah is not an English word, nor has Hannuka been ingrained into the English language far enough to have a set spelling, so you can spell Chanuka however you please.

As mentioned, I really have the most problem with the over commercialization with Christmas and it’s over penetration into society beyond what is reasonable, not with people saying Happy holidays or even Merry Christmas. The general assumption that everything Christmas related permeates all culture in America and by calling it a holiday celebration (with holiday tree and holiday mistletoe) that it is somehow now “neutral” with respect to religions is what is ridiculous.