First, because eggs can develop into chickens, chickens are carnivorous, so eggs are considered meat.

Second, eggs belong to birds and eggs in the menu. Birds are meat, so eggs are meat. They belong to the same category.

Third, the difference between vegetarian food and meat is that plant food is classified as vegetarian food, animal food as meat, and eggs are of course animal food, so they are meat. There is also evidence that vegetarian restaurants don't use eggs, so eggs are meat.