Do cats normally hunt squirrels? this is always a hot topic of discussion. Cats are natural predators; they're born to hunt. But here are some live pictures of cat hunting squirrels. An kitten playfully tosses a dead squirrel in air at a garden after killing it, in Mumbai on Monday. Your feline will probably chase anything that runs away, but the squirrel's bushy tail makes that particular rodent an enticing target. Tom may love chasing squirrels, but they can be hazardous to his health if he actually gets his paws on one.