1. Fill a pot with enough water to cover the eggs by at least an inch and bring it to a boil over high heat. Once boiling, reduce the heat to medium-high and cook for 6-8 minutes until the eggs are fully cooked.
2. Remove the eggs from the boiling water using a slotted spoon and transfer them to a bowl of ice water. Let the eggs sit in the ice water for at least 5 minutes to cool completely.
3. Peel the eggs and then cut them in half lengthwise. Remove the yolks and discard them if desired (optional).
4. In a large mixing bowl, combine the egg whites, mayonnaise, mustard, sweet relish, chopped onion, and chopped celery. Mix everything together until well combined.
5. Season with salt and pepper to taste. You can also add other seasonings such as paprika or dill if desired.
6. Serve the egg salad immediately or cover it and refrigerate it for at least 30 minutes before serving. This will allow the flavors to meld together and make it easier to slice.