He will not be going home after bootcamp, he will be going right to his school command. If the command is out of state odds are he will fly out the day after PIR.
The recruiter has given you and your SR wrong information. The only time that new Sailors get Leave after BC is when they have PIR during the Holiday Stand-down at the "A" Schools and your SR does not. He may have graduation from "A" School before Christmas if he classes up immediately and would be home for Christmas since MA "A" School is 7 weeks after classing up. If it takes a week or more to class up (right now it's a week), then he could come home sometime over the holiday stand-down, but not necessarily for Christmas and then return to Lackland to finish "A" School and then go straight to his first duty station after gradulation since he would have used up his Leave time.
It would be nice if recruiters actually kept up with how things happen at BC now instead of telling future Sailors and their loved ones stories that perhaps happened at one time (I think it has always only been those with PIR around Christmas and the New Year though) and getting their hopes up. (Some recruiters do stay informed, but many don't.)