"If I were a pixelated action hero, hard boiled eggs would be my medi-kits."
There are many ways to hard boil an egg, of course. Discovering what method you prefer is a matter of practice.
I fill the bottom of a pot with eggs (the less space they have to rattle around, the less liable they are to break during boiling), shake some salt over them, and cover them with water. I try to use water that is the same temperature as the eggs. I don't want to shock cold eggshells with hot water, even if it might make them come to a boil faster... the whole point of the process is to keep that fragile shell intact.
I bring to a boil, let boil five minutes, and let sit for 5-10 minutes before pouring cold water over them. You can skip the setting time, but your eggs will have dents in them. I let them sit in the cold water for a while before handling them -- because my fingers are important to me.
There is also a method of hard boiling known as "hard cooked" eggs. This involves bringing the eggs to a boil, and immediately taking them off the heat to set (15-20 minutes or so). The advantage of this is that you spare the eggs the knock-and-tumble of the boiling stage, once again preventing breakage.
How to peel a hard boiled egg:
Supposedly, the fresher the egg, the more difficult it will be to peel. When an egg is difficult to peel, you end up with a frustrating mess of rubbery eggy bits clustered around a rather lonely-looking yolk. It is very sad. That said, fresher eggs are also tastier eggs, so we must thwart the evil anti-peeling freshness forces.
Tap your cooked egg firmly on a hard surface several times, rotating it so that you crack all sides. Then, hold the egg between your palms and roll it back and forth to loosen the shell. Voila! The shell should come right off. If you still have problems, peel the egg under running water. And if that doesn't work, you still have a nice (if somewhat battered) treat for the cat.
The following is strictly a matter of opinion. A properly boiled egg, when just cooked, should have a texture like soft gelatin, with a velvety yolk. It should melt in your mouth. If the egg is undercooked, the yolk may be a bit moist, but still very good. However, if the egg is overcooked, it will be hard or rubbery, and the yolk will be very dry with a greenish exterior. I actually prefer hard boiled eggs after refrigeration. A properly boiled egg, when eaten cold, should have a creamy texture. And that, my children, is a wonderful thing.
I suppose this plants me firmly in the midst of the protein faction... d'ohwell...