Yes hard boiled eggs do need to be refrigerated if they are not gonna be eaten within 2 hours.
Hard boiled eggs are safe to be out of the refrigerator for up to 2 hours but then you should either eat the hard boiled or put them in the refrigerator within 2 hours.
If you're hard boiling eggs for Easter Egg hunts then wait to hard boil them and color them until you're ready for the egg hunting.
I hard boiled the eggs within an hour of the egg hunt and had them colored.
Then let the kids know that they need to be eaten or refrigerated if not gonna eat them right away.