Much discussion swirls around the best way to peel a hard-boiled egg. The older an egg, the larger the air cell, and the neater it peels. Thoroughly chilling hard-boiled eggs before peeling helps by firming the white. Holding an egg under a stream of water as you peel removes any bits of broken shell. Some cooks advocate a two-step method that involves plunging the cooked egg into ice water for two minutes, then reboiling it for exactly 30 seconds and peeling it immediately.
Higher heat overcooks the proteins, leaving the egg white tough and rubbery.
Overcooking may also cause a greenish black rim to form between the yolk and the white. This is harmless but yucky when it comes to gourmet cooking. We looked in several cookbooks, all of which said that when you take the eggs off the heat, you plunge them immediately in cold water to halt the cooking, as well as to help with the peeling. Of course, this doesn't work if you need to have the shell intact for whatever reason. It is true, too, that the eggs should be at least a few days old to be easiest to peel.