Kitty, kitty, kitty. . .
My cat gets rid of everything mouse-to-rabbit size in my house. He loves his work, is on duty 24/7, would resent being asked to take a vacation. Makes no noise, is litterbox trained, and as a bonus, you get a warm lapful of purrrrrr whenever you sit down to read the newspaper.
Treat him nice and he's yours forever. Don't get me wrong, it's not like a human/dog relationship, but the cat does realize on which side his bread is buttered. The 2 thing cats have over dogs as a pet is the ease of litterbox training, and that you can leave them home alone for a couple daysÑwith food, water, and litterboxÑand they don't wreck the place.
Show 'em an acceptable place or 2 to scratch and theyÑmostlyÑdon't scratch elsewhere. Declawing is an option but a hunting cat needs all its tools.
Got mine @ the local humane society. You want a young (not set in his ways) male (friendlier) neutered (if he never learns to spray to mark his territory he'll never be a problem) shorthair (less shedding). The local humane soc. always will have more cats/kittens than they can get homes for and so are eager for you to have them.
Don't let on to the humane soc. people that you will encourage its natural hunting instinctsÑthey want to be humane to pests too. They can't have it both ways but they won't/can't see that, and if you get 'em upset enough they won't give you a kitty. Just don't bring up the mousing/squirrelling item at allÑYou want a nice kitty to give a home to, who will purr on your lap while watching TV/reading the paper. You'll have to promise to get it neutered and have it vetted regularly, but you should do both anyway.
They will want a donation and it seems to me that's fair.
When it's all over everyone is happy. The humane people got a deserving cat a nice home. The cat gets a home, a job, and a person to share affection with. You get a pet and your house de-squirrelled. I fail to see a downside.