Prepare barbecue (medium heat). Place zucchini and crookneck squash on large baking sheet; brush all over with 3 tablespoons oil. Sprinkle with salt and pepper. Grill vegetables until tender and brown, turning occasionally, about 10 minutes. Transfer to plate and cool.
Cut vegetables diagonally into 1-inch-wide pieces. Place in large bowl. Add basil, cheese, balsamic vinegar, and remaining oil and toss to blend. Season to taste with salt and pepper and serve.