It's generally accepted that an all seed diet is unhealthy to a conure. Conures are omnivores, not herbivores. So an all-vegetable diet isn't prudent either. Eggs are an excellent source of protein and are rich in minerals, vitamins and essential amino acids. Additionally, eggs are one of the richest sources of choline.
Choline acts like a fat and cholesterol dissolver.
So, even though eggs contain a large amount of cholesterol, they also contain the ingredient that prevents the cholesterol from attaching itself to arterial walls. Seems conures are pretty smart about some foods. Given the chance though, a conure will eat an unhealthy all seed diet. For a conure, seeds have been compared to candy. They really won't resist them.