If you hunt other game, you understand the legalities of getting a license or permit and it may be the same when considering if you can eat squirrels from your backyard. Some places allow landowners to skip these requirements. However, we would suggest contacting your state DNR or department of conservation to verify your rights.
Even if there is a hunting season, you may still be able to hunt squirrels if they are nuisance animals. Many areas allow landowners to manage these issues, usually free or with just a nominal fee for a permit.