Zucchini has a high water content and it is best to drain some out before adding it to the other ingredients. After grating the zucchini put it into a colander with small holes or a mesh sieve, sprinkle a little salt on it and let it sit while you prepare the carrots and onion. Then just push the zucchini down and let the excess water drain out. Alternatively you can squeeze it with your hands.
You can make this a vegan meal by substituting the eggs for more chia seeds, 1 TBS/egg) that have been soaked in water for about 30 minutes or so. You can used ground chia or flax seeds as an egg substitute as well.