Simple, for both Happy Christmas and Merry Christmas it is: "Buon Natale!". Or, Merry Christmas and a Happy New Year is "Buon Natale e un Felice Anno Nuovo". Or, more generally, Happy Holidays is "Buone Feste".
An article about how to wish people a Merry Christmas in Italian, as well as other. It is no coincidence that Christmas Eve dinner in Italy is called Cenone, .