I've contemplated pickling eggs, and have researched recipes and methods, however it seems as though even if you can them after pickling, you can still only keep them for a few months. Since I don't know if we like pickled eggs anyway, I decided this wasn't the optimum method.
Then I read about dehydrating them. I even did a couple as an experiment. But I've been told repeatedly and LOUDLY that it's not safe to dehydrate eggs. And it takes a long time. So I guess I won't rely on that as my primary method either.
Of course, eggs do keep in the refrigerator for quite a good amount of time, as long as you don't wash them. Eggs have a coat of "bloom" on them that seals them from entering bacteria. Sometimes we get some pretty yucky eggs, but I give those to the dogs and only keep the relatively clean ones.
(By the way, the eggs that you buy at the grocery store are already 4-6 weeks old. From experience you'll know that you can keep them for quite some time after you've bought that carton of white eggs.)
So, my next plan is to freeze them. And while the freezer isn't my first choice for keeping anything (every winter some place in Oklahoma is without power for days or even weeks, it seems, from either ice storm or blizzard or some other disaster), some things are best kept there.
First I did the "float test" to weed out the too-old eggs. Put each egg in a deep pan of water. If it lays on the bottom of the pan it is a good, fresh egg. If one end stands up in the water but the other end is still on the bottom of the pan, it is not as fresh but still good. If the egg floats on top of the water, get rid of it carefully!
Then I broke a dozen eggs into a mixing bowl. I whisked the eggs and added 1/4 tsp of salt to each dozen (not to each egg!). I filled our ice cube trays with the whisked eggs and froze them. When frozen hard, I popped them out and stored in a gallon-size zip bag in the freezer. A dozen eggs fill more than one tray, so I use more than one cube as an egg "equivalent" - for instance, if I need 3 eggs I use 4 cubes.
When ready to use them, let them thaw in the refrigerator overnight and then add to a recipe or make an omelet or scrambled eggs. If you know you'll be using them in a baking recipe, you might want to add 1/4 tsp sugar instead of salt.
Putting Eggs By, Part One
Putting Eggs By, Part Two
Putting Eggs By, Part Three