Don's on to something. Instead of selecting a cat, let the mother nature select the cat. Cats have become domesticated and pretty much worthless. However, all are not created the same. Get a fresh dead rat (or squirrel), take to the animal shelter. Carry it past the adoptable cat's. One that snatches it away from you is what you're looking for. You may not find that cat immediately but when you do you'll know. Now your subsidize your predator twice/day and it'll hang around. The pests won't. Or if you despise cats, train a shortie.
No more squirrels.