In a covered 8- to 10-quart pan over high heat, bring beef, onion, garlic, and 1/2 cup water to a boil. Reduce heat to medium and simmer for 15 minutes, stirring occasionally. Uncover and cook over high heat, stirring often, until juices have evaporated and meat is browned, 10 to 20 minutes.
Add chili powder (use larger amount for spicier flavor) and cumin; stir until fragrant, about 30 seconds, then add broth and tomatoes, including juice. Cover and bring to a boil, then reduce heat and simmer until meat is almost tender to bite, about 1 hour.
Meanwhile, slice peel off squash and cut flesh into 3/4-inch cubes. Rinse celery and slice diagonally 1/4 inch thick.
Add squash, celery, and green chilies; cover and return to a boil over high heat. Reduce heat and simmer, covered, until squash and beef are tender when pierced, about 15 minutes longer. Stir in hominy and return to a boil. Skim and discard fat. Add salt and pepper to taste.
Serve soup from pan, or pour into a tureen. Put sour cream and cilantro in bowls and offer to add to taste.