Preheat the oven to 375 degrees. Grease a casserole dish with cooking spray.
Place the butternut squash noodles and broccoli (florets and spiralized stems, if using) in the casserole dish and sprinkle evenly over with quinoa. Set aside.
In a medium saucepan, whisk together the chicken broth and milk over medium-high heat. Once heated, add half of the cheese and whisk until the cheese melts. Remove from heat and pour the cheese mixture evenly over the vegetables in the casserole dish and season with garlic powder, onion powder, red pepper flakes, and salt and pepper. Stir together to combine.
Cover the casserole dish with foil and bake in the oven for 30 minutes at 375 degrees. Remove from the oven, stir, and sprinkle over the remaining cheese and cook uncovered for 10 minutes or until cheese melts.