Bring a large pot of lightly salted water to a boil. Cook lasagna noodles in the boiling water, stirring occasionally, until tender yet firm to the bite, about 8 minutes. Drain and transfer to paper towels; blot dry.
Heat oil in a large skillet over medium heat. Add mushrooms, onion, garlic, and Italian seasoning. Cook, stirring, until mushrooms are tender and any excess moisture has evaporated, about 15 minutes. Add tomato sauce, tomatoes, vinegar, 1/2 teaspoon salt, and crushed red pepper. Bring to a boil, then reduce heat and simmer, uncovered, about 8 minutes.
Preheat the oven to 375 degrees F (190 degrees C).
Stir together ricotta, egg, rosemary, black pepper, and remaining 1/2 teaspoon salt in a bowl.
Coat a 9×13-inch baking dish with cooking spray, then spread with 3/4 cup mushroom mixture. Layer on 3 noodles, half of the remaining mushroom mixture, and 2/3 cup shredded cheese. Repeat with 3 noodles, half of the ricotta mixture, and half of the squash. Then top with 3 noodles, remaining mushroom mixture, and 2/3 cup shredded cheese. Add remaining 3 noodles and remaining squash. Sprinkle with remaining 2/3 cup shredded cheese.
Bake, uncovered, until hot and bubbly, about 30 minutes. Let cool for 10 minutes. Slice into 12 pieces. Top with fresh sage leaves.