Foreign National Firearms Background Check Enhancement Act This bill requires a nonimmigrant alien to get Department of Justice (DOJ) approval before purchasing a firearm. Specifically, the party transferring the firearm to the alien must receive a copy of the DOJ waiver or confirm with the DOJ that such an alien falls within specified exceptions, such as the alien possessing a U.S. hunting license. Currently, a nonimmigrant alien may acquire a firearm if the alien qualifies for one of the specified exceptions without acquiring DOJ approval.