to test for freshness, put an egg on its side in a bowl of cold water. If it stays, it's fresh. If it inclines on an angle and is several days old. If it stands on end vertically it's over a week old. If the egg floats, very likely you have an egg on its way to being rotten.
select eggs that are 7-10 days old (above test), not fresh ones. The best way to boil an egg is to cook it about 20 min. at a slow simmer. Or you can start eggs in enough cold water to cover, bring just to a boil, turn heat off, cover with a lid and let eggs stand in the hot water for 20 min.
After cooking them (above) drain the eggs, lightly crack the pointy end, then immerse them in a big bowl of ice water until they are completely cold (use your own judgement). Placing eggs in the ice bath stablizes the iron in the yolk and the sulfur in the white so that they cannot combine to form ferrous sulfide (the unsightly but harmless dark-green coating on the outside of the yolk.