In a 3- to 4-quart pan over high heat, bring about 1 1/2 quarts water to a boil. Peel chayotes and cut into 1/2-inch cubes (soft seed is edible). Add chayotes to water, and simmer until tender when pierced, about 10 minutes. Drain. Immerse in cold water until cool; drain again.
Meanwhile, stem and seed chili and bell pepper. Finely chop chili. Cut bell pepper into 1/4-inch squares. Cut cheese into 1/4- by 1-inch sticks. Peel and pit avocado and cut into 1/2-inch cubes. Combine ingredients in a wide bowl.
In a small bowl, mix vinegar, oil, and cilantro. Add vinegar mixture, chayotes, and onions to vegetables in wide bowl. Mix gently. Add salt and pepper to taste.