In a covered 3- to 4-quart pan over high heat, bring 5 cups broth and chili to a boil.
Meanwhile, peel chayote and cut into 1/2-inch cubes (include edible seed). Gently rinse and drain blossoms. Leave baby squash stems (female flowers); if desired, trim off fuzzy stems (male flowers). Set aside 6 of the prettiest blossoms; chop the remainder.
Add chayote to boiling broth. Mix masa flour with remaining 1 cup broth and stir into pan. Stir until boiling. Reduce heat to medium-low, cover, simmer, and stir occasionally, until chayote is tender when pierced, about 10 minutes.
Stir corn, spinach, and chopped squash blossoms into soup, turn heat to high, and cook until corn is hot, about 2 minutes.
Ladle soup into bowls and lay 1 blossom in each. Add cheese, salt, and pepper to taste.