Hill stepped into the fray which had been sparked between teammate Richard Dunne and a fan who was hurling X-rated abuse in front of one young fan.
The Daily Mai says Hill's issue with the supporter was in relation to the language he was using in front of the youngster who had accompanied him to Burnley.
The former Stoke and Crystal Palace defender, in no uncertain terms, questioned the example the furious fan was setting the youngster with his swear word rant.
The child supporter is said to have looked 'petrified' by the scenes that unfolded as the adult supporter used a string of expletives to get his views across.
Joey Barton helped to de-escalate the situation and prevented Leroy Fer from going into the crowd, though it is understood the Dutchman just wanted to give his shirt to a fan as he does after every game.
However, it is understood the ill-tempered situation continued outside the ground as other Rangers fans, furious at his behaviour, confronted the supporter in question.
It is claimed there were altercations between the angry fan and other members of the travelling support, which are said to have turned physical.