I submit to you photographic proof that your friend is wrong. A fried egg *can* be put on a burger.
but don't just take google's image search for granted.
Tyler Florence, celebrity chef, puts fried egg on his "Le Grande" burger at his restaurant, Wayfare Tavern.
Chef Laurent Tourondel (not so famous, but a french-trained chef, who earned his bonafides at Saint Vincent Ecole de Cuisine in Montlucon) puts a fried egg on his burger at BLT Burger at The Mirage. The BLT stands for Bistro Laurent Tourondel, not (as I originally thought) Bacon Lettuce and Tomato. I've eaten at a BLT Steak, and I can tell you Chef Tourondel's food is top notch.
But don't take the French's word for it. Gordon Ramsey's BurGR has a burger with fried egg on it.
The burger chain, Red Robin, offers fried egg on their burgers.
In the Jewish tradition, eggs are considered
pareve(neither dairy nor meat), and therefore may be placed on a burger.
In the Muslim tradition, meat and eggs are both Halal if properly prepared, and there is no restriction on being served together in a tasty fashion under one bun.
If you are a Buddhist or Hindu that practices vegetarianism, you certainly wouldn't put an egg on your burger--because you wouldn't be having a burger in the first place.
In a strange and sad twist, if you were a Seventh Day Adventist, you could have the egg, but not the burger. Perhaps they could make due with a big ol' grilled portabella cap. That sounds pretty delicious now that I think about it. Put a little sprouts and avocado. A tangy garlic aioli. Yum.
I can go on and on. But the bottom line is, your friend is simply wrong.