This gets discussed each year this time.
To summarize:
No serious modern classical scholar still holds the theory that Christmas was created to replace or ride on the back of an existing pagan celebration. There were no other Pagan religious celebrations on December 25, with the possible exception of a Sol Invictus celebration that was a minor one, and which appeared long after the rise of Christianity. The foremost expert on the Sol Invictus cult in the world discounts the idea that Christmas was created to ride on Sol Invictus’ coattails.
No other messiah or god figure was said to be born on December 25 in classical times. I can tell you the source of your information, daddygirl, and it’s been thoroughly debunked. It was from a book written by a Spiritualist who cited no sources for any of his claims. No classical scholar takes that idea seriously, but you can still find the book on atheist and anti-Catholic websites.
The whole “no sheep in winter” idea came from a Scottish anti-Catholic (Hislop). Sheep may not be outside in the Scottish winter, but they sure are in Bethlehem. Also in Navajo country, where I live and where the sheep are out all year around. We are on the same latitude line and elevation as Bethlehem.
The Solstice as an astronomical event (known and used as a marker in all agricultural cultures) as opposed to a religious one seems to be the best explanation for the choice to commemorate the Feast of the Nativity, as Augustine said (and oddly, the cults of Apollo and Sol Invictus did not use any solar events such as solstice or equinox dates as religious holidays).
The early Christians believed that the mixture or dilution of Christian worship with pagan worship was so sinful that they went to their deaths rather than accept it, so it is hardly plausible that they would combine their faith with a pagan one; after all, the early Christians (and the Jews) believed that the pagan “gods” were false gods not because they didn’t exist, but that they did exist and were demons and unclean spirits.
The specific choice of 12/25 could also be based on a calculation based on the conception dates of Elizabeth, the presumed conception date of Mary (if we assume that she conceived shortly after the Annunciation), and the dates when Zecheriah’s priestly line offered sacrifices in the Temple. Plug all that in, and you have a date for birth in late December. I would think that people in the early Church know what the human gestation period is and can do simple math, so that may have led them to the 12/25 date.