“Happy New Years” means you are wishing the person happiness in all the New Years to come, as well as happy Chinese New Year, Jewish New Year, etc. etc. It covers every conceivable New Year Celebration for that person for the foreseeable future and wishes them only happy ones.
I prefer to say “Merry New Year” in a fake African accent a la Eddie Murphy in Trading Places.
I have never encountered “Happy New Years” (note lack of apostrophe).
What I hear most commonly, and would say myself, is “Happy New Year” (said like this: “Hap-py New Year!”): the wish is that the whole new year, not just one day of it, would be happy.
But I don’t see anything wrong with telling someone to “have a happy New Year’s” (i.e. New Year’s Day and/or Eve).
I probably say both. I live in the land of Kroger’s, Meijer’s and a bunch of other places that have no 's in their official name, but are constantly referred to as if they did. I’ve always figured “Happy New Year’s” was a related phenomenon.
“Happy New Year.” One. There’ll be another one next year; don’t rush it.
Or “New Year’s Eve,” or “New Year’s Day” (possessive, not plural).
In our December 1999 concert, my chorus performed the song, “What are You Doing New Year’s Eve.” We had a hell of a time getting the Second Tenors to sing the line “Welcoming in the new year, New Year’s Eve” correctly.