Preheat the oven to 350°F. Spray a 9x5-inch loaf pan with baking spray, or line it with parchment paper and spray with baking spray, set aside. In a medium-sized bowl, stir together the flour, baking soda, baking powder, cinnamon, nutmeg, and salt until combined, set aside.
In a large bowl cream together the butter and sugar for 2 minutes until combined. Add the egg and mix it in. Mix in the rum extract and vanilla. Add the dry ingredients to the wet and mix it in until just combined. Add the eggnog and fully mix it in until there are no lumps.
Add the batter to the prepared pan. Bake for 50 to 55 minutes until a toothpick inserted into the center comes out clean. If the bread looks like it is browning too soon, tent it with foil about halfway through. Let cool in the pan for 10 minutes. Take the bread out of the pan and let it cool on a wire rack.
While the bread cools, make the glaze by whisking together the powdered sugar and eggnog in a medium-sized bowl until smooth. The glaze should be thick yet pourable.
Place the bread on a wire rack over a piece of parchment paper. Pour the glaze over the bread and let it drip down the sides. Let it set for 15 minutes before slicing.