Although food safety sites recommend refrigerating eggs, eggs in their shells have natural chemical and physical protection from the bacteria whose growth would be inhibited by refrigeration. In many parts of the world, eggs are not routinely refrigerated. Now, from a liability standpoint, I can’t recommend that you store your eggs on the counter, since there is a very small chance (like 1 in 20,000) that an egg with an intact shell might still harbor bacteria inside of it. But I will tell you that you can. And my mom, who spent a decade living on a small sailboat with a tiny
refrigerator would tell you that they’ll stay fresher if you turn them
over daily. Warm eggs will scramble up fluffier (find the same fluffiness by taking them out of the fridge the night before, or soaking them in warm water for a few minutes before cracking them.) And one more thing: older eggs are better for hard-boiled eggs. The moisture loss creates air space between the membrane and the shell that makes them easier to peel.