Toss together first 8 ingredients in a large bowl.
Grill vegetables, covered with grill lid, over medium-high heat (350° to 400°), turning occasionally, 6 minutes or until they are tender.
Cut zucchini and squash halves into thin slices, and place in a large bowl. Add remaining grilled vegetables, pasta, and basil, tossing gently; sprinkle with cheese.
To hold the onions together on the grill, insert a wooden pick through each slice.