Heat coals or gas grill for direct heat. Spray half of one side of four 18x12-inch sheets of heavy-duty aluminum foil with cooking spray.
Mix mushrooms, green beans, bell pepper strips, onion and barbecue sauce. Place vegetable mixture on sprayed side of foil sheets. Top with burger patties. Fold foil over patties and vegetables so edges meet. Seal edges, making tight 1/2-inch fold; fold again. Allow space on sides for circulation and expansion.
Cover and grill packets 4 to 6 inches from medium heat 20 to 25 minutes or until vegetables are tender. Place packets on plates; unfold foil.