There is no solid biblical proof of any date--whether it be Dec. 25th, September/October, spring or any other popular guess. But, here are a couple of items for consideration.
The following is a link to the Quarterly Statement - Palestine Exploration Fund:
https://bit.ly/2Hyjyad
This document provides a table of mean temperatures in Palestine between the years 1845-1854. The mean temperature in December during these years was 61.5, January's mean was 57.3 and February's 58.6. Apparently the Middle East has enjoyed mild winters for a long time, except in mountainous regions.
And following is a quote from The Life and Times of Jesus the Messiah by Alfred Edersheim, a devout Jew who accepted Jesus as his Savior. Originally written in the 1800's, this quote is from pp. 131-132 of the new one-volume complete and unabridged edition:
"There is no adequate reason for questioning the historical accuracy of [December 25th]. The objections generally made rest on grounds which seem to me historically untenable...But a curious piece of evidence comes to us from a Jewish source. In the addition to the Megillath Taanith (ed. Warsh. p. 20) the 9th Tebheth is marked as a fast day, and it is added, but the reason for this addition is not stated. Now, Jewish chronologists have fixed on that day as that of Christ's birth, and it is remarkable that, between the years of 500 and 816 A.D. the 25th of December fell no less than twelve times on the 9th Tebheth. If the 9th Tebheth, 25th December, was regarded as the birthday of Christ we can understand the concealment about it." (p 132)
"And yet Jewish tradition may here prove both illustrative and helpful. That the Messiah was to be born in Bethlehem, was a settled conviction. Equally so, was the belief , that He was to be revealed from Migdal Eder, 'the tower of the flock.' This Migdal Eder was not the watchtower for the ordinary flocks which pastured on the barren sheepground beyond Bethlehem, but lay close to the town, on the road to Jerusalem. A passage in the Mishnah leads to the conclusion, that the flocks, which pastured there, were destined for Temple-sacrifices, and, accordingly, that the shepherds, who watched over them, were not ordinary shepherds...The same Mishnaic passage also leads us to infer, that these flocks lay out all the year round, since they are spoken of as in the fields thirty days before the Passover -- that is, in the month of February, when in Palestine the average rainfall is nearly greatest. Thus, Jewish tradition in some dim manner apprehended the first revelation of the Messiah from that Migdal Eder, where shepherds watched the Temple-flocks all the year round. Of the deep symbolic significance of such a coincidence, it is needless to speak". (p 131)