Preheat oven to 450°F. To a medium microwave-safe bowl, add the butternut squash and 3 tbsp water. Season with salt and pepper. Cover with vented plastic and microwave 8–10 min., until tender.
Remove plastic from butternut squash and let cool. Using a fork, mash squash until smooth. Add the black beans and lightly mash. Season with salt and pepper.
Line a large baking sheet with parchment paper and coat with cooking spray. Arrange 8 tortillas on baking sheet. Spread squash mixture evenly over tortillas and top with the cotija cheese. Top with remaining tortillas, pressing down to adhere. Coat tops of tortillas generously with cooking spray.
Bake 15–20 min., until quesadillas are golden brown and crisp around edges, flipping once halfway through. Meanwhile, microwave the rice according to package directions. Cut quesadillas into halves and serve with rice.