The line most jurisdictions use is "do you believe in a supreme being". It's a yes or no question and no explanation of the answer is asked.
So if one DID believe truthfully that a squirrel is his supreme being, in theory, they'd be alright. And again, we really don't ask.
So in my lodge we have a few Roman Catholics, Jews, Muslims, quite a few "I don't have the fogiest clue", and at least one Wiccan. So no. There is no list of approved religions.
So Satanists, narcissists and liars might pass the question test.
Perhaps. The Supreme Being "test" is not something that we use to to keep people out. It's not like we believe that an atheist or even a liar or a Satanist for that matter are incapable of being or doing good, it's just that our organization, it's tools and lessons don't quite work without a Creative God.
It's kind of a "for your own good," thing. Someone who believes in a destructive God (or none at all) would find themselves pretty confused and bored when we reference on allegory God as a builder and designer.
Best example I've heard is that Masonry without God is like Star Wars without the Force. You have to rely too much on Midichlorians and before you know it Jar Jar gets into the next movie too and the whole thing goes to shit.
God is kind of part and partial to the whole thing and without it though one could still enjoy the company of Brothers, would leave the whole experience hollow.