Preheat oven to 350 degrees F (175 degrees C). Place squash, cut-side down, in a baking dish; add water.

Bake in the preheated oven until squash is tender, about 30 minutes.

Heat a large skillet over medium-high heat. Cook and stir chorizo in the hot skillet until browned and crumbly, 5 to 7 minutes; drain and discard grease.

Heat olive oil in a separate skillet over medium heat; cook and stir onion and garlic until onion is softened and browned, 10 to 15 minutes. Add green bell pepper and cook until softened, 3 to 4 minutes. Add tomato, chicken stock, black beans, and corn; cook and stir until liquid is evaporated, about 10 minutes. Stir chorizo into mixture.

Flip squash over and fill each half with chorizo mixture; top with Cheddar cheese.

Bake in the preheated oven until cheese is melted, about 5 minutes.