Three things:
1) The manager, based on the available information, sounds terrible at customer relations and inventory management. It's one egg and not a dozen; if the restaurant is unable to provide one egg to a loyal customer -- for fear of running out of eggs -- then the manager needs to re-engage with their training on customer and order satisfaction.
2) While the customer was absolutely not entitled to the egg, he/she should have been provided the egg on the burger because denying the customer the egg is denying the customer an egg. Regardless of the customer's attitude towards the staff, denying someone an egg is like denying someone a mint. It's an insignificant item that most people can offer others at no significant or extra drain on their resources, so there's literally no way the restaurant can come out looking like good guys by denying someone a single egg.
3) If I wanted an order to be modified, and the restaurant said "Nah bro", I would simply recognize that the restaurant can't satisfy that particular order. If I want to eat something else then I'll go back to the restaurant, but if I want that special order, then I'll go to a restaurant that can deliver it. It's like buying anything else. For example, if I want a 30 inch IPS monitor that can do 4K HDR USB-C G-sync G-unit etc. for under $500, and LG isn't able to provide that, then I'll go to a manufacturer that can. If I'm in the market for a large French-door refrigerator for under $5000, and LG is able to provide it, then I'll go to LG and buy it. Being loyal to a brand is, for roughly 98% of the time, a waste of your resources.