DRAIN beets, reserving juice. Set beets aside for another use. COMBINE beet juice, vinegar, sugar and spices in medium saucepan. Bring to a boil; STIR until sugar is dissolved. REDUCE heat and simmer 5 minutes.
ARRANGE eggs in 1-quart heat-proof glass jar with tight fitting lid. POUR hot marinade over eggs. Cover tightly. Allow to cool to room temperature 1 hour. Refrigerate to blend flavors, at least several hours or up to one week.
Easy 12-Minute Method for Hard-Boiled Eggs: Place eggs in a saucepan large enough to hold them in a single layer. Add cold water to cover the eggs by 1 inch. Heat over high heat just to boiling. Remove from the burner. Cover pan. Let eggs stand in hot water for about 12 minutes for large eggs (9 minutes for medium eggs; 15 minutes for extra large eggs). Drain. Shock the eggs in a bowl of ice water to cool them immediately. Hard-boiled eggs are easiest to peel right after cooling.