Line the bottom of the prepared baking dish with 1/2 of the bread cubes. Sprinkle with ham, Cheddar cheese, and Swiss cheese, and top with remaining bread.
Beat together eggs, milk, and onion powder in a large bowl. Pour evenly over bread. Cover and refrigerate overnight.
Preheat the oven to 375 degrees F (190 degrees C).
Blend crushed cornflakes and melted butter in a small bowl. Spread evenly over casserole.
Bake in the preheated oven until bubbly and golden brown, about 40 minutes. Let stand 5 minutes before serving.