The short answer
Yes — plain cooked egg is a great occasional protein boost. Never feed it raw.
Why
Cooked egg is an excellent source of protein and amino acids, especially helpful during moulting or breeding. It must be cooked (hard-boiled or plain-scrambled with no oil, salt or milk); raw egg carries a salmonella risk.
How to feed it
Offer a small piece of hard-boiled or plain-scrambled egg and remove it after a couple of hours. You can also crush and bake the eggshell and sprinkle it on food for a calcium boost.
How much & how often
A small amount 1–2 times a week.