For the high caloric price of admission you get very little vitamins, minerals, and protein. Just look at their nutrient profile for proof: 1 egg nets you 10+% of your daily saturated fat recommendation, 14% riboflavin, 11% vitamin b12, and all other vitamins less than 8%. For the minerals, 23% of your selenium (really not something you need in abundance - google selenium overdose), 10% phosphorus, and the rest 5% or below. All of that for only about 1/9th of your daily suggested protein.
So to get your daily protein from eggs you would need to overdose on cholesterol by almost 9 times the recommendation, consume 90+% of your saturated fat, and overdose on selenium, and so on.
And if that weren’t enough, you would be eating more fat (by weight) than protein.
Edit-fixed saturated fat content to 10%
Edit 2-The ratio of omega3 to omega6 is about 1:12. That should be more like 1:1 (preferably) or at least 1:4. So to correct that you would need to eat even more fats to correct that ratio intake.