I coached high school football for years and the rule was that if your mouthguard was not in during play it was a penalty.
Even if you did the "fishhook" with your mouthguard it was a penalty.

I see players all over the place with their mouthguards hanging out. Research shows mouthguards can not only save your teeth but be helpful in minimizing concussions, so why is it OK for everyone to have their mouthguards hanging out.

With all the safety issues in football I don;t understand why they don;t make them keep the mouthguards in.