It should be a meat dish, for three reasons:

First, because eggs can develop into chickens, and chickens are meat, so eggs are also considered meat dishes.

Second, eggs belong to the poultry category in recipes. Poultry is a meat dish, so eggs are also meat dishes, and they belong to the same category.

Third, the distinction between meat and vegetables is to classify plant foods as vegetarian dishes and animal foods as meat dishes. Of course, eggs are animal foods, so they are meat dishes. There is also evidence that vegetarian restaurants do not use eggs, so eggs are meat dishes.