Preheat the oven to 400°F (200°C). Toss together butternut squash, 2 teaspoons of the oil, 1/2 teaspoon of the salt, and 1/4 teaspoon of the pepper. Arrange in a single layer on a rimmed baking sheet.

Bake in the preheated oven until softened and lightly browned, 15 to 20 minutes. Set aside.

Meanwhile, heat remaining 2 teaspoons of oil in a large skillet over medium-high heat. Add onion and 1/4 teaspoon of the salt, and cook, stirring often, until softened and translucent, 3 to 4 minutes. Add spinach, and cook, stirring often, until wilted, 1 to 2 minutes. Remove from heat, and set aside.

Combine shredded provolone and mozzarella in a bowl. Stir together ricotta, parsley, eggs, 1 1/2 cups of the shredded cheese mixture, and remaining 3/4 teaspoon salt and 1/2 teaspoon pepper in a large bowl.

Lightly coat a 13x9-inch baking dish with cooking spray. Spread 1 cup of the marinara on the bottom. Arrange 4 noodles side by side over sauce; spread half of ricotta mixture over noodles. Spread roasted squash evenly over ricotta mixture; top with 1 1/2 cups marinara. Top with 4 more noodles and remaining ricotta mixture. Spread spinach mixture evenly over ricotta mixture; top with 1 1/2 cups marinara. Top with remaining 4 noodles and 1 cup marinara. Sprinkle with Parmesan and remaining shredded cheese mixture. Cover with aluminum foil.