Because most of the holidays that Western cultures celebrate aren't celebrated in Japan, there is no direct translation for "happy holidays" in Japanese. However, katakana allows Japanese speakers to approximate phrases for holiday greetings with a Japanese pronunciation.
For example, "Merry Christmas" turns into "merii kurisumasu" in katakana. "Happy Hanukah" in katakana is "happii hanuka", and Happy Kwanzaa is "happii kuwanza". These all approximate the American pronunciations but in a way that fits into Japanese writing and speech. New Year's Day is an important day in Japanese culture, and therefore the greeting "Happy New Year" does have a direct translation: Akemasjite omedetoo gozaimasu.