There is more than one way to boil an egg. Some people swear by poking a pinhole in the top of an egg before boiling. Others say to bring the water to a boil first, and then add the eggs.
I consulted two hard-boiled experts, my mother and grandmother. This is how they cook them:
Place eggs in pan, cover with cool water.
Bring water to a gentle boil. Cover and remove from heat.
Let sit for 12-15minutes.
Plunge cooked eggs in ice water to cool.
I found this egg peeling tip on www.deviledeggs.com (yes, there is a website just for deviled eggs). Once the egg has cooled, gently crack it all around, and then reemerge it in the cold water for a few minutes. The shell should come off easily.