Preheat oven to 350 degrees. Grease and flour a 9 inch square baking pan.
In a large bowl, beat together the eggs and sugar until creamy. Stir in the vegetable oil until well combined. Add the grated carrots, orange juice and walnuts and mix well. Add the flour, baking powder, salt, cinnamon and nutmeg and stir until just combined. Pour into prepared baking pan and bake for 25 - 30 minutes or until a toothpick inserted in the center comes out clean.