If they are in the country legally with a green card, and meet all the same background requirements as citizens, yes.
Someone who has not proven themselves to be a threat, and has broken no laws, I believe should be guaranteed the same second amendment right as the rest of us.
The right to self defense is a basic human right after-all, and until a person shows by their own actions that they cannot be trusted, they should be able to exercise that right.
If the person is in the country illegally however, hell no!
I make no assumptions about the circumstances that prompted an individual to sneak into this country, but the fact remains they have already committed a crime by being here in the manner they chose to enter.
There is a legal process, they chose to ignore it, so what other legal process will they choose to ignore while they are here? Illegals have every expectation to get away with anything and simply be deported as an ultimate result, short of perhaps cold blooded murder.