Hi movinon,
I am not aware of any law barring a non citizen from possesing a firearm, assuming the person has a clean record. But I'm not sure, having never dealt with the situation. I suggest you Google BATF and contact someone if you want to know definitively.
However, if he has any convictions for a felony, or has been convicted of domestic violence (even a misdemeanor) or has a restraining order against him regarding a domestic partner, been judged mentally defective, is an illegal drug abuser or has been dishonorably discharged from the military, then he would be generally prohibited from possesing a firearm.
As you point out though he could simply store it at a friends house, although this would subject his friend to possible prosecution if your nut is prohibited from possesing a firearm.
Hope that helps.
mud