Preheat oven to 350 degrees.
Combine flour, baking soda, and cinnamon in a large bowl.
In another large bowl, beat together eggs, oil, applesauce, vanilla and white and brown sugars.
Pour dry ingredients into the bowl with the wet ingredients. Mix until a batter forms.
Fold shredded zucchini into the batter until it is evenly distributed.
Grease a muffin tin with cooking spray. Pour batter into muffins tins, filling each one to the top.
Bake for 25 minutes at 350 degrees or until toothpick inserted in center comes out clean.