I spent a lot of years in the restaurant biz as a server/bartender/some management, and my opinion is that the GM is totally in the wrong in the story. Just put an egg on his goddamn burger, especially if he's a heavy regular. What on earth, this shouldn't even be a question.
The family of regional restaurants that I worked for really prided themselves on customer service, though. We had one family of regulars who, no kidding, had one child who wouldn't eat our kids menu, and wanted a happy meal from the McDonalds across the street. Every time they came in, we sent a busser or the host across the crosswalk to get him one. We were proud to do it, because we were giving the parents the chance to go out and have a steak or whatever on a semi regular basis, and if we didn't do it, they wouldn't have that chance. A couple of years later, the kid had outgrown his pickiness, and would order off our menu, and they were still regulars at that point. It was a proud anecdote that was always explained to new hires when they came on.
Refusing to put an egg on a customer's burger. That's nuts.