Q: What breed lays easy-to-peel eggs?
A: If you're losing a lot of egg white when you try to peel hard-boiled eggs, the problem is not related to the breed you have, so choose your favorite!
What causes the difficulty peeling is the freshness of your eggs, believe it or not. Fresh eggs are difficult to peel, and the eggs you are getting at home now are very fresh indeed. The problem--if having fresh eggs can be regarded as a problem!--is that what peels easily are old eggs. The eggs you were getting at the grocery store were very old, sometimes a month or more old. As eggs age, some of the moisture from inside the egg evaporates, so there is more room between the membrane and the cooked egg, resulting in an easier to peel egg when it is boiled. So, when I make deviled eggs, for example, I always use my oldest eggs. For egg salad, appearance doesn't matter quite so much, but it will still be an easier experience if the eggs you use are a couple weeks old rather than trying to use this morning's freshest eggs! Use your freshest for sunny side up, or over easy... or baking, or poaching, or sharing... the problem is keeping those delicious eggs around long enough that you have some older eggs to boil.
So remember, if you want your eggs to peel more easily when hard-boiled, use your oldest eggs for that purpose. Which breed laid the eggs doesn't matter.
Another thing you can do is add a little baking soda to the water you use to cook your eggs; use a teaspoon or so in your water. This isn't something I like to do, because it makes the eggs smell extra sulfury... but if you're desperate, it may be something you want to try. Eggs cooked this way are easier to peel. And here's one last tip about hard boiled eggs: don't actually boil them. Boiling the eggs means you are more likely to get those unsightly green rings around the yolks.
The green yolk rings happen because when your egg gets too hot, the iron in the yolk reacts with the sulfur in the white to produce an unappetizing, greenish grey substance called ferrous sulfide. That substance turns your beautiful orange yolks very ugly! Not to mention that overcooking hard boiled eggs can make the whites rubbery and tough, too.
In our house, to get that "perfect" hard boiled egg, we start our eggs in cold water in a pan and bring just to a good boil--then we immediately turn the heat off. Leave the eggs in the hot water; they will continue to cook! The exact time you want to leave them in the hot water will depend on the size of your eggs. Small, bantam eggs may get done more quickly than larger eggs. Since we're usually cooking large eggs at our house, it takes 20 minutes or so for them to get done. When your time is up, plunge your eggs in ice water to stop the cooking fast. Change the ice water a few times to make sure they are chilled thoroughly, then peel.
Voila: now you have perfect hard boiled eggs, easy to peel and with tender whites and beautiful yolks.