Heat olive oil in a large skillet over medium heat; cook and stir onion, red bell pepper, mushrooms, and garlic in the hot oil until vegetables are tender, about 10 minutes.
Mix tomatoes with green chile peppers, black beans, salsa, chili powder, cumin, cayenne pepper, salt, and black pepper into onion mixture; simmer, stirring occasionally, until cooked through, about 5 minutes. Add spinach; cover skillet and simmer until spinach turns bright green, 1 to 2 minutes.
Spread squash into an 8x8-inch casserole dish; top with tomato mixture. Sprinkle crackers over tomato mixture.
Bake in the preheated oven until crackers are browned, 35 to 40 minutes. Sprinkle casserole with Cheddar cheese.