What I want to know is how to hard-boil eggs so that they’ll peel cleanly every time, without the white pulling off on the shell. Tried every cooking technique I’ve heard, but none works with every egg every time. My wife even got one of those air-pressure gadgets once, but it went in the trash after ruining 5 of a test batch of 6.
Also, I put the eggs in cold water (with inch on top) and then bring to a boil (as opposed to putting eggs directly in boiling water). After boiling, replace the hot water with cold water and let sit for a while.
When peeling, roll the eggs gently but firmly over a cutting board or other hard surface until many cracks are produced around the middle (equator) of the egg. Then peel a narrow strip with your fingernail across the "equator" of the egg. You should find that the top and bottom parts of the shell slide right off.
Sounds a little complicated but really doesn't add too much time. I use a sewing needle to make the pin-prick in the egg.
No one knows why. Eggs just are not all the same. And I do like deviled eggs, so I get ticked if they look gnarly...
I use a steam cooker now, but this will work with a pot of water as well:
As soon as they are done (still very hot), VERY QUICKLY put all of them in ice water. Prepare a pot of ice water early, with ice cubes having time to get the water cold. Have it ready - no delay.
After a few minutes (2 or 3 does it for me) - take the eggs straight out of the ice water and tap them on the side by putting them firmly on a hard surface, cracking the shell. ROLL the egg under your hand on the clean, hard surface, fracturing a large part of the shell. Squeeze a bit in your hand, and begin to peel. I find it often comes off in two or three big pieces.
OH - just re-read your question. If boiling in a pot, add some salt to get the water to boil hotter. Watch for the first real bubbles, and don’t let them boil past 10 minutes.

“hard-boil eggs so that they’ll peel cleanly every time”
For me, this works:
Put the eggs in a pan and cover with cold water.
Bring to a boil uncovered — about 15 minutes for me.
Turn burner to lowest heat, and cover the pan.
Some turn burner completely off but I don’t.
Set timer for 15 minutes.
Drain. Add cold water and a lot of ice.
Shell should slide right off in one piece.