The solstice isn't always on the 21st December, sometimes it can be either side. The date is not set in stone, it is the date of the physical day with the shortest hours of sunlight that defines the date of the solstice that year.
As has been said by many already, our ancient people were more in tune with sun and moon cycles then most of us are. For more on that ignore the pagan books and read the proper archaeological ones instead

For me it is not the Green Man who is reborn, the Green Man is not a God, in fact I have never heard reference to the Green Man being born at this time, he is for me,an extremely different being altogether

For me it is the God of the sun and of the land, the God who is all male, all fertile who is reborn at this time

Obviously Yule predates xmas, and also xmas did not used to be celebrated at this time of year either, Jesus was not born in December, there has been much evidence to show this.
And again I'm afraid, that saying Merry Christmas rather than Happy Christmas has no Pagan conotactions, "Merry Meet and Merry Part" is only a few decades old. Merry means, merry, having a good time, a cheerful time and is for me, nicer than a happy time