Bring 4 quarts water to a boil in a large Dutch oven; add macaroni, and cook 4 minutes.
Add sliced carrot, and cook 2 minutes; add broccoli flowerets, and cook 2 minutes. Stir in squash, and cook 1 minute; drain.
Combine cottage cheese and next 5 ingredients in a large bowl; stir in pasta mixture, mushrooms, mozzarella cheese, and half of Cheddar cheese. Spoon into a lightly greased 2-quart baking dish; sprinkle with Parmesan cheese and remaining Cheddar cheese.
Bake at 375° for 20 minutes, shielding with foil after 15 minutes to prevent overbrowning if necessary.