Everyone should not be required to own a gun, but everyone should *be able to* own a gun. The right to bear arms is protected by the Constitution, and the Constitution applies to all US citizens on US territory, not just non-felons. We don't strip perjurers of their right to free speech, so why do so many people think it is okay to strip arms rights from violent felons? Both have demonstrated irresponsibility with a particular tool, but why the double standard? They get convicted, serve their time, and should be released as free men, not as sub-citizens who lose part of their constitutional protection.
I don't know* whether the Constitution applies to illegal aliens or not, but if it does I think it should apply 100% - not just partially. The document was not designed to allow rulers to pick and choose which sections apply and which do not.
* I suppose the question of whether it applies is based on which conditions you must meet for it to apply. Obviously the Constitution only applies on US territory. So given that a person is on US territory, does that person have to be a US citizen to be protected by the constitution, or does just occupying a spot on US land afford a person constitutional protection?