I don't have any ideas about why they're getting bitten by squirrels so often but I do have a suggestion to reduce the number of vet visits needed.
You can buy Wellness Treats from the vending machine at the default vet clinic in Brindleton Bay as long as it's unowned. The Wellness Treats should also be available at any other unowned vet clinics, as long as they have one of the vending machines that came with Cats and Dogs. You can make them yourself if you have level 5 Veterinarian skill. The Wellness Treats seem to cure all illnesses, so just stock up on them whenever you are at the vet and then use them as needed.