Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.
Heat olive oil in a large skillet over medium heat. Add onion and garlic; cook until onion is translucent, about 5 minutes.
Stir in zucchini, yellow squash, and bell pepper; cook, stirring frequently, until zucchini and yellow squash are tender, about 10 minutes.
Stir in black beans, green salsa, cumin, and cayenne pepper; cook and stir until thickened, 5 to 8 minutes.
Divide 1 package Mexican-style cheese among tortillas. Spoon zucchini filling on top cheese in a line across middle of each tortilla.
Fold opposing edges of tortillas to overlap the filling. Roll 1 of the opposing edges around the filling creating a burrito; place, seam-sides down, into the prepared baking dish.
Bake in the preheated oven until cheese is melted and burritos are heated through, about 15 minutes.
Sprinkle remaining 1 package cheese over burritos before serving.
