Heat oil in a large skillet. Add onion, green pepper, and carrot. Cook for 5 minutes, until onion is softened. Stir in broth, tomato sauce, brown sugar, white vinegar, pineapple with juice, and ketchup. Simmer for 10 minutes, or until vegetables are tender.
Meanwhile, soak bread in milk in a large bowl until milk is absorbed. Mix together with ground chicken, shredded zucchini, egg, salt, black pepper, and garlic powder. Shape into 12 meatballs, about 2 tablespoons each.
Drop meatballs into simmering sauce. Cook 15 minutes, or until cooked through, turn over once during cooking.
Remove skillet from heat. Stir in rice. Cover, and let stand 5 minutes. Fluff rice with fork before serving.