I thought Army Airborne school was only like three weeks long. And it's very likely the boyfriend will be sent somewhere else afterwards. Is she going to move to Italy for three weeks?
Even when he reaches his first "permanent" assignment he will not be able to live off base while single unless he is promoted to probably an E-6. If they marry then they will likely be assigned family housing on base, or given housing pay and be able to live off the base if the on base housing is full.
But, even if he is married, the Army will not let him bring his wife along to live with him during training. They will pay him a housing allowance for his family dependents, since the Army requires that all members provide housing for their dependents, but she will be staying at home during his training.
She could be a camp follower living in motels as he bounces from station to station if she doesn't want to get married, provided she can afford it. His pay won't cover it, and he'll still be spending his nights in his assigned barracks on base.
I'd have your friend read the housing section of "What the Recruiter Never Told You."
http://usmilitary.about.com/cs/joiningup/a/military6.htm