Preheat oven to 350°F. Prepare two 8-inch round cake pans with butter and flour or use non-stick baking spray.
Grate a large zucchini (or two small zucchini) and place in a clean dish towel. Life the edges of the towel up and squeeze some of the extra moisture out, but you do not want the zucchini to be dry. You will want 2 total cups of shredded zucchini after it has been drained. Set aside.
In a large bowl and using a hand mixer, beat together the eggs, oil, vanilla, and sugar. Fold in the zucchini.
Slowly stir in the flour, salt, baking powder, and baking soda. Gently fold in the blueberries. Divide batter evenly between prepared cake pans.
Bake 35-40 minutes in the preheated oven, or until a knife inserted in the center of a cake comes out clean. Cool 20 minutes in pans, then turn out onto wire racks to cool completely.