So, yes 22-250 is certainly legal as long as it is shooting expanding bullets.Rifles and shotguns
Utah Code § 23-20-3 and Utah Admin. Rule R657-5-8
You may use a rifle or shotgun to take big game, but your firearm and ammunition must meet the following requirements:
•Your rifle must fire centerfire cartridges and expanding bullets.
•Your shotgun must be 20 gauge or larger, and you may use only slug ammunition or buckshot that's 00 or larger in size.