1. Place rack in upper third of oven. Preheat to 400°.
2. Heat a grill pan over high heat. Lightly coat pan with cooking spray. Add naan; grill 1 minute on each side or until grill marks appear. Place naan on a baking sheet. Add onions to pan; grill 2 minutes, turning once.
3. Rub cut sides of garlic over top side of naan. Drizzle with 2 teaspoons oil. Layer squash over naan, leaving a 1/2-inch border. Sprinkle corn over squash; top with onions. Sprinkle with 1/4 teaspoon pepper and salt; top with cheeses. Bake at 400° for 8 minutes. Turn broiler on (do not remove flatbreads from oven); broil 2 minutes or until lightly browned. Sprinkle with remaining 2 teaspoons oil, remaining 1/4 teaspoon pepper, and thyme. Cut each flatbread in half.