They are covered starting the first day of basic, on I-day once they in-process.
As for taking them off your private insurance, that will depend on your insurance carrier. Some, you just call and tell them your young adult has joined the military, and they will remove them. Other companies required proof of new coverage, which your son/daughter can get, but they will not have the time to access the MyCare/TriCare website and download the letter during Basic. So, the soonest you might get this documented proof would be after A-Day.