I found this online:
Storing of hard-cooked (boiled) eggs:
Refrigeration is necessary for hard boiled eggs if the eggs are not to be consumed within a few hours.
It is preferable not to peel your eggs until you are ready to eat or use in your recipe. Hard-cooked eggs in the shell can be refrigerated up to one (1) week.
Peeled hard-boiled eggs can be stored in the refrigerator in a bowl of cold water to cover for about 1 week (change the water daily) - or in a sealed container without water (cover the eggs with damp paper towels) for the same length of time.
SAFETY NOTE: It is not safe to leave hard-boiled eggs (including those in their shells) out at room temperature for long. If they have been taken to a picnic, or served on a buffet, keep them cool (in a cooler) while they are being served, and discard the leftovers.