Yep, just plain old hard boiled eggs.
I've struggled with this for years. Mine never peeled nicely no matter what I did. I added salt and vinegar to the water. I boiled them for different lengths of time. I started them in cold water and hot. I tried really fresh eggs and not so fresh eggs. Nothing seemed to make much difference.
Until I moved to Washington. The water here is much softer than our Utah water was. Even with a water softener there and no water softener here, our water is softer now. And my eggs almost always peel without problems.
This really isn't that important of a problem, unless you like deviled eggs and want to serve some that don't look like mutant eggs of a spawning demon.
So here are the tricks and tips I've found to help you boil the perfect eggs.
Hard Boiled Eggs
1/4 c. vinegar
1-6 t. salt
Fresher eggs really are better but not too fresh. If you buy them at the store, use them within a week for the best results. If you get them straight from your own chickens, refrigerate them for 3-5 days before using for hard-boiled eggs.
Put your eggs in a largish pot. You want enough room for them to wiggle a little but not too much. Cover them with room temperature water. You want about half an inch of water on top of the eggs. Add vinegar. Add in salt. Go with 1-2 t. if your water is fairly soft, go with more if you have hard water. If you don't know, be safe and use about 3 t. of salt. Regular table salt works just fine for this.
Bring to boil over high heat. Turn heat to low, cover the pot, and let the eggs simmer for 16-20 minutes. Altitude makes a difference. Lower altitudes need shorter times, higher ones longer cooking time. Preference matters, too. If you like your eggs completely hard, cook them for the longer time. If you like your yolks bright yellow and just barely set, cook the eggs for the shorter time.
When the eggs are done, drain the hot water off. Stick the pot in the sink and turn the faucet on to cold. Let it pour over the eggs, dumping out the pot and refilling it several times. You want the eggs to cool down as fast as possible. Let the eggs soak in cold water for a few minutes, changing the water if it gets warm at all. You can even add ice to the water if you want.
Skip this next step if you're coloring the eggs for Easter. Cracked eggs let the color inside the shell and you get colored egg whites.
After about 30 minutes, pour off the water. Gently crack the shells by dropping the eggs on top of each other or stirring them around the pot. You want lots of small cracks. Fill the pot with more cold water and let the eggs soak for another 5-10 minutes. You want water to get inside the shell at this point.
Pull out an egg. Leave the rest in the cold water. Gently roll the egg on the counter to crack the shell the rest of the way. The shell should separate and pull right off. There is a membrane just inside the shell. If you can get your thumb inside that, the shell slides right off the eggs.
Repeat with the rest of the eggs. You should have pretty, smooth, perfectly peeled eggs. If you don't, you can always chop them up for potato salad or egg salad or creamed eggs. Or just make some ugly spawn-of-alien-demon deviled eggs.