Check to see whether the eggs have any cracks. Boil water in a pot. If the eggs leak, quickly add some vinegar to stiffen the egg whites. (Adjust the amount to the size of your pot.)
To prevent cracking, use a ladle or similar implement to gently immerse each egg into the boiling water. Boil to your desired degree of hardness.
Transfer to chilled water and allow to cool to the center. Once cool, remove the shell by peeling off the membrane. Even with half-boiled eggs, the shell comes right off.
If you first put the eggs in the pot, then pour boiling water over them from an electric water kettle or other source, then this will shorten the time it takes to boil the eggs.
Story Behind this Recipe
One day, when I needed to boil an egg, I put them in with some vegetables I was boiling, and I was amazed. The shells came right off! This is just one of my little theories, and I feel like I owe an apology to my economics teacher. This time, I had 5 successes and 1 failure. But, if you're going to use them chopped, even if a little egg white leaks out, at least the shell comes right off!
There is a chance the eggs may crack since they are put in boiling water. For recipes where the shape of the egg is important, boil a little more water than you need. The shock of the difference in temperature will make the shells come right off. Immerse the eggs as gently as possible. For broken eggs, mash them with a whisk or chop them up.