For perfect hard-cooked eggs, place raw eggs in a saucepan large enough to hold them in a single layer. Add enough water to cover by 1 inch; heat over medium-high heat until the water is almost boiling. When one or two bubbles break on the surface, remove the pan from the heat and cover. Let stand 15 minutes for large eggs. Drain and rinse under cold running water. Unpeeled eggs can be kept in the refrigerator for up to 1 week.
The cooked vegetables, dressing, shrimp and hard-cooked eggs can be prepared ahead. Refrigerate and assemble just before serving.