I used to have the hardest time getting decent hard boiled eggs. They'd be green, or not done, or wouldn't peel right or....occasionally would be perfect. One day, a woman that I work with was eating a hard boiled egg. It was lovely. I asked her how she boils her egg. She said, "I don't know, I just boil them. They always come out." Yikes. An idiot savant of the hard boiled egg!
So, I went home, and figured it out.
The last part first. The key to easy peeling is...tada...old eggs. Not REALLY old eggs, but not fresh from the chicken. If you buy AA eggs you are going to have trouble peeling them. So I buy eggs by the multiple dozen, particularly when they are on sale, and when they've been in the refrigerator for about a week, they can be hard boiled and will peel pretty easily.
As for the hard boiling? Don't boil them...hard. Here's the method/recipe
Put eggs in a saucepan that has a lid, with enough water to cover by at least an inch. Bring to a boil. As SOON as the water comes to a boil (not a simmer, a full boil) take off the heat, cover with the lid, and SET THE TIMER for 11 minutes. When the timer goes off, pour off the hot water, and with the eggs still in the pan, stream in cold water for a couple of minutes. Finally, dump a bunch of ice on top and let them sit for about 10 minutes.
The eggs will be perfect. Really.