Illegal batting or not?
 

A is in shotgun formation on their own 10-yard line. Snap goes way over QB's head into the end zone. QB turns and chases the ball and bats it backwards out of the end zone over the end line. Is this illegal batting and, if so, what's the enforcement? Please cite the NF rule(s) or case(s).

Illegal Batting
 

Yes, it is illegal batting. Enforcement would be from the spot of the foul, and thusly, a safety. Since the ball went out of the endzone it'll be a safety whether it's accepted or not.

NFHS is 9-7-2 I think.

NCAA has slightly more liberal batting rules. In NCAA you cannot bat forwards in the field of play and cannot bat in any direction in the end zone (9-4-1-c).