excellent question
Seems to me that if you think they should be able to get driver's licenses and health insurance and what not then you pretty much have to think they should be able to own guns. If you don't think they should have those things then you have to think they should not be allowed to own guns. But I could be convinced otherwise by a good argument I think.
Personally I'm torn it, leaning to the "no, they can't" side. There are very good arguments on both sides.
There are strong policy arguments for illegals having driver's licenses, and certainly health insurance (e.g., infectious disease control), that do not exist when it comes to gun ownership. The best policy argument that I can think of, assuming everybody must register the guns they own (which I strongly favor), is that more guns used in crimes will be traceable than would be the case without gun registration, and that would apply to illegal aliens owning guns as well.
Putting aside their status crime, is there any evidence that illegals are more likely to commit crimes, than those illegally here of similar SES status? I tend to doubt that, but I am curious to know the answer. If not, than that argument would not be as play as much as a policy reason to deny guns to illegals.