Preheat the oven to 350 degrees F (175 degrees C). Coat an 11x7-inch baking pan with butter.

Beat eggs in a bowl until frothy. Add sugar and continue to beat until thick and creamy, 4 to 5 minutes. Add rum and vanilla extract. Add flour, baking powder, and salt and mix until just combined; do not overbeat. Pour batter into the prepared baking pan.

Bake in the preheated oven until a toothpick inserted in the center comes out clean, 20 to 23 minutes. Allow to cool.

Combine sweetened condensed milk, eggnog, and rum in a bowl and stir until well combined. Poke holes throughout the cooled cake using a skewer or toothpick. Slowly pour sauce over the cake. Cover with aluminum foil and refrigerate until sauce is absorbed, at least 2 hours.