Place the eggs in a saucepan and cover them with about 1 inch of cold water. Bring the water to a boil, cover the saucepan, remove it from the heat, and let the eggs stand for 10 to 12 minutes.

Transfer the cooked eggs to a bowl of ice water and cool them completely for 20 to 30 minutes. Peel the eggs carefully, rinsing away any small shell pieces.

Drain the canned beets while reserving 1 cup of the beet liquid. If the beets are whole, cut them into slices or wedges.

Combine the beet juice, white vinegar, sugar, salt, optional peppercorns, and sliced onion in a saucepan. Bring the mixture just to a simmer, stirring until the sugar and salt dissolve; do not let it boil vigorously.

Arrange the peeled eggs and beets in a clean glass jar or other nonreactive container. Pour the warm pickling liquid over everything, making sure the eggs are mostly submerged.

Let the container cool briefly, then cover it and refrigerate. Keep the eggs chilled while they pickle.

Refrigerate for at least 24 hours before eating. For stronger flavor and a more intense pink color, wait 48 to 72 hours.