Preheat the oven to 325 degrees F (165 degrees C). Grease a 9x13-inch baking dish.
Beat eggs in a large bowl with an electric mixer until fluffy. Beat in sugar, oil, and vanilla; gradually mix in flour, baking powder, cinnamon, and nutmeg until combined. Fold in squash and pour batter into the prepared baking dish.
Bake in the preheated oven until a knife inserted into the center of cake comes out clean, about 45 minutes.