The .22-250 (and the Swift) are intended for long distance varminting on prairie dogs, jackrabbits, feral domestic pets and coyotes. Mostly.
Since cottontails and gray squirrels are considered eatin'-meat--small game, as noted--it's best to consider the plain old .22 rimfire so as not to destroy the little critters.
Some states require hunting licenses for squirrels, and there may well be seasons. Also, many states have laws about wasting game meat.
Regards,
Art