Preheat oven to 350 degrees F (175 degrees C). Grease bottom only of a 9x5 inch loaf pan, or three 6x3 inch loaf pans.
Beat eggs in large bowl. Stir in sugar, melted butter, eggnog, rum extract, and vanilla. Combine the flour, baking powder, salt, and nutmeg. Stir into eggnog mixture, just enough to moisten dry ingredients. Pour batter into prepared pan or pans.
Bake bread in large pan for 40 to 60 minutes, or until a toothpick inserted in the center comes out clean. Breads baked in the smaller pans require 25 to 40 minutes. Let cool in pan for 10 minutes, then turn out onto a wire rack, and cool completely. Wrap tightly, and store in the refrigerator.