You don't know that. Some say December, some say April due to sbeepherding time, you say June. Who gives a fuck really? Jesus' actual birth date isn't really known.
“The more I study science the more I believe in God.” -- Albert Einstein
According to your story, his parents were on the way from Jerusalem where they were for the 2 weeks of High Holidays (Rosh Hashana + 10 days of atonement + Yom Kippur). When they were traveling back Mary gave birth. High Holidays always fall in September so the birth should be somewhere ether late September or early October. The only thing that happened on December 25th was Pagan Winter Solstice