Preheat oven to 350°F (175°C). Grease a 9x5-inch loaf pan or line with parchment paper.
In a bowl, whisk together the eggs, sugar, applesauce, and vanilla extract.
In another bowl, mix the flour, baking soda, baking powder, salt, cinnamon, and nutmeg.
Gradually combine the wet and dry ingredients.
Gently fold in the shredded zucchini and chopped strawberries.
Pour the batter into the prepared loaf pan.
Bake for 55–65 minutes, or until a toothpick inserted in the center comes out clean.
Allow the bread to cool for 10 minutes in the pan, then remove and place on a wire rack to cool completely.