Heat oil in a stockpot over medium heat; add garlic and cook until fragrant, about 30 seconds. Add onions; cook and stir until translucent, about 5 minutes. Stir tomatoes into onion mixture and cook for 30 seconds more.
Mix cousa squash and chicken bouillon into tomato mixture; reduce heat, cover, and simmer until squash is tender, 20 to 30 minutes. Stir in corn and Colby-Monterey Jack cheese.