Place eggs in single layer in large saucepan or Dutch oven; add enough cold water to cover eggs by 1 inch. Bring to a boil. Immediately remove from heat; cover and let stand 15 minutes. Drain; rinse with cold water. Place eggs in bowl of ice water; let stand 10 minutes. Peel eggs; cut eggs into quarters. Set aside.
In medium saucepan, prepare sauce mix with milk and butter as directed on package. Stir in quartered hard-cooked eggs, lemon juice, dill and pepper. Cook and stir 2 to 3 minutes or until thoroughly heated.
Place ham on microwave-safe plate; cover loosely with microwave-safe plastic wrap. Microwave on Medium for 2 to 3 minutes or until thoroughly heated.
Place 2 muffin halves on each serving plate. Arrange ham slices on muffins. Spoon egg mixture over ham.