Did he have a right?
It depends…if he’s authorized, or deputized, then he probably has a license to protect. If he felt that person was endangering his clients, then its probably legal. To the bodyguards knowledge, that person could have a weapon or is trying to do harm to his clients. And given that his clients are famous and with children, the person shouldn’t be approaching them in the first place.

What if it was you in their position? Would you want a stranger coming up to you, when you have your children with you?

Yes, the bodyguard’s duty is to guard bodies! Tom Cruise was squirted with a pistol filled with water, but it could very well have been filled with acid.