It's "Happy New Year," not "Happy New Years."

You wish people a Happy New Year, not a Happy New Years. It is neither plural nor possessive and earns no letter "s," with or without apostrophe. You hope people have a good time on New Year’s Eve, and also on New Year’s Day, but when saying, "Happy New Year," such well wishing expresses your hope for an entire year’s worth of good fortune. There are a few possessive holidays, and they include Valentine’s Day, Mother’s and Father’s Day, and if you’re a fan of Henry V, St. Crispin’s Day. On a related note, Presidents’ Day is plural possessive and Veterans Day is plural. As to the capitalization of "Happy New Year," convention and AP hold that all three letters should be capitalized.