I always use the oldest eggs for hard boiling. I run them under cold water right after boiling and they usually peel ok. Someone said add vinegar to the water maybe? Hmm I'll go try to find where I read something like that...
This is the answer I got a few years back when I had the same question. You can peel fresh hard boiled eggs if you cook them like this. Bring enough water to cover your eggs to a boil. Gently use a large spoon to place each egg in the boiling water. Boil 14 minutes. Remove from boiling water and submerge in enough cold water with ice cubes (needs to be very cold). Wait until the eggs have cooled down, crack all around and peel. Shell should come off without ripping the white.
I have tried fresh eggs and month-old eggs. Couldn't tell a difference.
I have not tried putting eggs into water that is already boiling. And I've always cheated and used cold tap water (well water), but never ice cubes.
I haven't tried salt or vinegar in the boiling water. Vinegar makes sense, as it would start to decalcify the shell.
I've been poaching eggs for egg salad, but my kids love deviled eggs and I've always wanted to try the hard boiled eggs wrapped in sausage. Scottish egg?
A trick for peeling fresh eggs my MIL taught me that works but you don't always get a perfectly smooth egg: Crack the cool, hard-boiled egg lightly all over, then using a spoon carefully sandwiched between the egg and the shell so that the spoon is conforming to the shape of the egg, gradually loosen the shell from the egg.
This is the only way I've found to boil eggs that are easy to peel. Place eggs in a pot with cold water to cover. Bring to a boil & boil for 5 minutes. Using a potato masher or wooden spoon, gently crack the egg shells under the boiling water. Boil for another 5 minutes. Place in ice-cold water until chilled. The shells should come off easily. Your timing may vary according to your altitude.