Place the eggs in a small saucepan and cover with water. Bring to a boil over medium heat. As soon as the water is boiling, remove from the heat and cover. Let eggs sit for 15 minutes.
Drain the water and run cold water over the eggs to cool.
When the eggs are cold, peel and chop them. Place eggs into a medium bowl.
Add the mayonnaise, dill, chives, horseradish, mustard, vinegar, salt, and pepper to the bowl with the eggs and stir to combine.
Spread the mixture over the bagels and serve immediately.
NOTE: Double the egg salad mixture for a thicker topping on each bagel.