Squirrels...just MAYBE with a perfect headshot. Rabbits...I wouldn't even try it. You'll probably just wound the animal, leaving it to die slowly days later.

500 FPS is not enough power, in my opinion, for anything other than plinking and target practice, certainly not enough power for making a clean kill while hunting.
I'd look for something around 800-or higher FPS if you want to hunt with an air rifle. I've had good luck with Gamo air rifles, and I DO NOT recommend Beeman, (see my post in the 'do not buy this' thread.) I use .177 round-nose pellets for hunting/plinking. No experience with .22 pellets.