The worst thing you can do for an egg that is about to be boiled is to start it in cold water. Bring your eggs from the fridge cold to boiling with the water causes the membrane to merge with the white, resulting in stained and marked devil’s eggs. (And no, I never found the vinegar trick useful.)
I like to start my eggs (pitted) in boiling water, let them boil for 10 minutes, then put them in an ice bath (if there is a whole bunch of them) or rinse them in cold water. (if there are only a few and I plan to eat them immediately). I rarely have trouble peeling. I would say I come across a slightly unpeelable egg every dozen or so, which is acceptable considering that eggs and their shells are made up of complex organ systems.
You can also steam your eggs for similar effects (and very tender whites). Simply fill a pot with about an inch or so of water, place a steamer basket inside, cover and bring to a boil over high heat. Add the eggs and cook for 11-12 minutes before cooling in an ice bath or under cold running water.