If you are sending somebody a greeting, capitalise thus:
Merry Christmas and a Happy New Year.
Within a narrative text, we write 'I had a merry Christmas, all right - I was drunk all the time' (Christmas is always capitalised).
If New Year refers to the festival, it is capitalised -
'We had a great New Year party';
but not if it refers to a simple calendar date -
'I intend to drink less in the new year'.