Wait, let me guess... As such, they are not allowed to buy weapons, right? How convenient...
Seriously: I fully support background checks -- extending what we already have, to cover all sales. I also think that if motor vehicles require registration, insurance, and license to operate, then firearms sure as heck should, too.
Even so, if I'm being intellectually honest, I also have a twinge of doubt when it comes to "mental health" grounds, and I have some concern about a Minority Report pre-cog sort of exercise.
This will probably just get me down-votes from both sides of the debate, but, there you have it.