Im wondering what gauge wire to use for snaring rabbits. I have 20 gauge, do you think that is big enough?

That's plenty for bunnies .

I'd recommend cable over wire. Wire will do it, but cable is much stronger per size. You can buy cable at the hardware store in 1/16", which will do, but is even bigger than you need. They should also have roles of picture hanger cable wich is finer yet, but plenty strong.