How should "Merry Christmas" and "Happy New Year" be capitalized?

How should "Merry Christmas" and "Happy New Year" be capitalized?

Here are some examples taken from some of the top results on Google:

I’d like to be among the first to wish you a very Merry Christmas.

Share the spirit of peace and joy on this occasion of Merry Christmas with your acquaintances and loved ones.

Just wanted to wish you all a Merry Christmas, Happy Holidays, or a Happy whatever you are celebrating at this time of year.

And here is what Wiktionary says:

Merry Christmas and a Happy New Year!

Is this capitalization correct? If so, does this mean that "Merry Christmas" and "Happy New Year" are proper nouns?

As others have mentioned, Christmas and New Year are proper nouns, and thus are capitalized. Generally the phrases "Merry Christmas" and "Happy New Year" are used in greetings, as headings, or in some other isolated way, and thus "Happy" and "Merry" are the first word of the sentence, and thus those words are capitalized.

Happy New Year!

is a sentence by itself, and thus Happy should be capitalized. It would not be necessary to capitalize "birthday" if you were saying "Happy birthday" instead of "Happy New Year".

I wish you a merry Christmas and happy New Year.

is how I'd capitalize the words if they weren't being used on their own, but rather in a longer sentence.

Christmas and New Year are proper nouns.

I know that Merry Christmas and other such greetings are usually capitalised, but the reason why is unclear.

They are not proper nouns, but they are phrases and as such the first letter of the first word is often capitalised, but this is not set in stone. I get the impression that their are cultural differences, that people state-side are more likely not to capitalise Merry and Happy, but I can't confirm this.

You could also perhaps consider them quotes that have lost the quotation marks:

We wish you a 'Merry Christmas'

However, there is a school of thought that goes against this; they consider merry Christmas perfectly acceptable, though to me is looks rather odd.