Hey John,
I have the occasion to enforce the hunting laws. I looked through the ODNR rules and regs and can't find anything limiting you from using a 22LR handgun for squirrel. The only quirk would be on State Park areas where they have different rules. So I think your game warden was misinformed or they changed the rule since you had contact with him. You can check the 2018-2019 hunting regs online from ODNR, but I can't find it and I've hunted squirrel with my Ruger MKII, no problems.
Mike