Combine first 3 ingredients in a large bowl, stirring until blended. Press crumb mixture into bottom of a lightly greased 8" or 9" square pan. Bake at 350° for 8 minutes. Cool.
Beat cream cheese, sugar, and rum extract at medium speed with an electric mixer just until smooth. Add eggs, 1 at a time, beating just until blended. Pour 1 1/2 cups cream cheese batter evenly over baked crust.
Stir eggnog into remaining batter. Divide batter in half; stir melted white chocolate into 1 portion. Spoon evenly over cream cheese batter. Stir melted semisweet chocolate into remaining batter. Drop spoonfuls of chocolate batter evenly over white chocolate layer; gently swirl batters with a knife.
Bake at 350° for 30 to 33 minutes or until almost set. Cool completely on a wire rack. Cover and chill at least 1 hour before serving. Cut into bars.