Qualifications
As we've already discussed, what kind of security job you're talking about will dictate what the qualifications are. Maybe the better way to put it is this way: You can ask for whatever job you want, but your skill set will determine how likely you are to get it. Sort of like asking the IRS to change its mind about auditing you. They'll first check your qualifications (are you a tax-law writing Congressperson?), and then respond accordingly ("No!").
If you're applying for a mall cop job, having a pulse and being able to tie your shoes will get you most of the way there. Employers do look for one or two other things, though. Since the job has to do with crime prevention, it's probably best not to have a felony conviction.
That's the case with pretty much every job, come to think of it, unless you're applying for the newly vacant position of Prison Tattoo Design Coordinator. Then it might be a plus.
Anyway, your entry-level security guard jobs aren't too demanding in terms of qualifications. A few basic ones you can expect are high school degree or equivalent, or, if you don't have that, several years of traceable work history, a driver's license (because sometimes they work late hours and not all pizza joints deliver), permission to work in the U.S. if not a citizen, that kind of thing (source).
If you're aiming for the Blackwater-type stuff and you're not a former Army Ranger, Green Beret, Navy SEAL, Marine, cop, FBI, or something of the kind, it will be a little tougher to get in. Those outfits are small and tight knit, the kind of groups where everyone knows each other, and your high-end private security companies often employ through the Bro Network.
Guys who got out of the service and now work there talk to their buddies who are still active and try to recruit them. Your high-dollar, high-risk security companies are looking for things like military service (Special Forces preferred), prior law enforcement, foreign language ability, intelligence community experience, and the like.
But again, these options are kind of the extremes. There's everything in between out there, too.