First of all, it is certain that eggs are definitely not "meat dishes". The real meat dishes in Buddhism refer to warm and seasoned vegetables, that is, garlic, leeks, onions and canals (as if they were like lilies). Buddhists think that eating it will make people lust, so they don't eat it The meat we usually talk about is "fishy" food (that's how fishy meat comes from). Buddhists avoid killing, so they avoid fishy dishes.
Eggs depend on whether they are fertilized or not. A fertilized egg should be able to hatch chicks, so it is alive and therefore fishy. The unfertilized eggs can't hatch chickens, and they don't kill if they eat them, so they are vegetarian.
But people who don't eat any part of animals, including eggs and milk.
In addition, it is definitely wrong to mention that "protein is a meat dish" (meat dishes here should refer to fishy dishes).
Protein is an essential nutrient for human body. If you don't take it for a long time, it won't be long before you die of malnutrition (remember the big-headed doll incident that made a lot of noise a while ago, because those children were short of milk powder from protein for a long time).
Monks and people who fast for a long time don't lack protein, because many vegetables also contain protein. For example, beans are the most abundant plants with protein, and their protein is only slightly inferior to meat.