The refrigerator is where you should store your eggs. Unless you seldom open the door, it's best to place the eggs on an inside shelf. Repeated opening and closing of the door causes temperature fluctuations and slamming can result in breakage. The carton in which you purchase them helps keep the eggs from picking up odors and flavors from other foods and helps prevent moisture loss—a particularly important factor if you have a frost-free refrigerator.
Fresh uncooked eggs in the shell can be kept refrigerated in their cartons for at least 4 to 5 weeks beyond the pack date. Properly handled and stored, eggs rarely "spoil". If you keep them long enough, they are more likely to simply dry up! But, don't leave eggs out. They'll age more in 1 day at room temperature than they will in 1 week in the refrigerator.
As soon as you've cooled them, refrigerate hard-cooked eggs in their shells in their cartons and use them within 1 week.
When storing hard-cooked eggs, you may notice a "gassy" odor in your refrigerator. It may be more noticeable when the refrigerator is opened infrequently. The odor is caused by hydrogen sulfide which forms when the eggs are cooked, is harmless and usually dissipates within a few hours.
For outdoor eating occasions, eggs can be kept refrigerator-cold with ice or commercial coolant in an insulated bag or picnic cooler as long as the ice lasts or the coolant remains almost at freezing. Unless it's quite cold weather, for hiking, backpacking, camping and boating when refrigeration or cooler facilities aren't available, it's better to use dried eggs. Usually available in sporting goods stores, dried eggs can be reconstituted with purified water and used in most of the ways you would use fresh eggs. Specially coated hard-cooked eggs which keep without refrigeration for a considerable length of time are also available in some areas. Pickling and other forms of preservation are additional possibilities.
If a recipe calls for only whites or only yolks, refrigerate the leftover whites in a covered container up to 4 days. Store yolks in water in a covered container in the refrigerator and use in a day or 2. If you can't use the yolks quickly enough, hard cook them. Carefully place them in a single layer in a saucepan and add enough water to come at least 1 inch above the yolks. Cover and quickly bring just to boiling. Remove from heat and let stand, covered, in the hot water for about 15 minutes. Remove with a slotted spoon and store in a tightly sealed container in the refrigerator up to 4 or 5 days.
If you find yourself with more eggs than you will use in several weeks, freeze them.