Make stock: Pull meat off bones and dice enough to make 2 cups; set aside. Put bones in a 6-qt. pot and pour in 9 cups water (to just cover bones). Add celery, carrot, onion quarter, and bay leaf. Bring to a boil, then reduce heat and simmer, lid barely ajar, 2 hours. Strain stock and return to cleaned pot.
Add lime juice, salt to taste, chopped onion, zucchini, and corn. Bring to a simmer, then add 2 cups diced reserved chicken. Simmer until vegetables are just cooked, 1 to 2 minutes more. Taste and season with salt.
Put 1/3 cup rice in each of 4 bowls. Ladle soup over rice. Stem and halve squash blossoms (if using) and add to bowls. Top with some cheese and a grind of pepper.
MAKE AHEAD Soup through Step 2, up to 1 day. Cooked rice, up to 2 days.