Place water, meat, onion, and garlic in a large pot. Bring to a boil and then reduce heat to low, cover and cook for about 4 hours, or until meat is fall off the bone tender.
Discard onion and garlic and continue to simmer the meat and broth
Meanwhile, place the chiles in a small pot and cover them with water. Simmer until chiles are softened, about 4-5 minutes
Place chiles and their water in a blender and blend until smooth (with the chiles and the broth in the blender, I had about 3.5 cups). Using a colander, strain the chile mixture into the soup, discarding the pulp
Add in corn, chayotes, and potatoes and continue to simmer covered about 1 hour, until just beginning to soften
Add in the green beans, squash, and epazote leaves. Continue cooking until the vegetables are tender, about another 40-45 minutes
Add salt to taste and serve immediately