Cut 2 (15- x 13-inch) rectangles of parchment paper; fold each rectangle in half lengthwise. Place parchment sheets on an ungreased baking sheet, and open out flat.
Place 1 fillet on half of each parchment sheet near the crease. Place 6 tomato halves, cut side down, on one side of each fillet; place squash and zucchini slices alternately on other side of each fillet. Sprinkle fish with green onions, cheese, salt, and pepper. Drizzle dressing over fish.
Fold paper edges over to seal securely; cut off open corners to make a half-moon. Starting with rounded edge of fold, pleat and crimp edges of parchment to make an airtight seal. Bake at 400° for 15 minutes or until packets are puffed.
To serve, place packets on individual serving plates. Cut an opening in the top of each packet, and fold paper back. Serve immediately.