Combine softened ice cream and rum; stir until blended. Spoon mixture into ice cube trays. (You should be able to make about 28 ice cream cubes.) Freeze at least 8 hours. (Ice cream cubes will not freeze hard like ice. You'll be able to remove them as soft cubes.)
Fill individual glasses two-thirds full with eggnog. Add 2 or 3 ice cream cubes to each glass. Sprinkle with nutmeg. Serve immediately.
Note: You can combine the eggnog and ice cream cubes in a punch bowl and serve with a ladle, if desired.