Yes — cooked egg is excellent protein for parrots, especially during molt, and crushed cooked shell adds calcium. Serve it plain and fully cooked.
What it means for conures
There's nothing specific to a conure that changes the verdict on eggs — it's treated the same as for parrots generally. A conure can handle a moderate portion.
The full picture on eggs
Egg is the gold-standard protein boost for molting, breeding, or recovering birds — complete amino acids plus vitamin B12 and choline. Yes, it's a bird eating egg; nutritionally it's no different from the insects and occasional eggs wild parrots opportunistically eat. The rules: fully cooked (raw egg risks salmonella), and completely plain.
Hard-boiled wedges or dry-pan scrambled, cooled.
A teaspoon-sized portion for small birds, more for macaws, once or twice a week.
Rinse and bake shells (10 min at 120°C) before crushing them in.