I'm partial to the lever actions for .22's. But they would be above your desired budget ($250 for the basic Henry levergun, new).

Have you considered an air rifle? You can get a very nice one in your price range. You can take squirrels and rabbits with these easily. Ammo is dirt cheap, the most commonly found being .22 and .177 caliber. They can be dead-on accurate if you buy a reasonable one.