Preheat oven to 350 degrees F (175 degrees C). Grease one 9x13 inch baking pan.
Stir together the pumpkin, banana, egg, and egg whites in a large bowl. Beat in maple syrup, sugar, and vanilla; the batter should be a bit frothy. Stir in zucchini; set aside.
Mix flour, bran, salt, baking soda, baking powder, cinnamon, nutmeg, cloves, ginger, and allspice in a separate bowl. Gradually add the flour mixture to the zucchini mixture, stirring just to moisten all ingredients. Over-mixing the batter will make it tough.
Pour the batter into the prepared pan. Bake until a toothpick inserted in the center comes out clean, 50 to 60 minutes. Cool completely before cutting into squares.