Here’s an easy way to see if your eggs are fresh: Fill a bowl with water, and carefully drop in the eggs. If they sink, they are fresh and still just fine to eat. (Congratulations! You should make a Tortilla Espanola.) If your eggs float and bob at the surface, they should be tossed. (Sad times.)
Very fresh eggs have less air in them, causing them to sink in water, and making them harder to peel when hard boiled. The eggs that bob at the bottom of your bowl of water, not sinking completely but not floating at the top, are actually perfect for hard boiling (and turning into Deviled Eggs). That little bit of air just inside the shell will make the task of peeling a heck of a lot easier.