Preheat oven to 425°F; line a baking sheet with parchment. Cut squash in half lengthwise. Discard seeds and membranes. Rub cut sides of squash with 1 tablespoon oil. Place squash, cut sides down, on baking sheet. Bake until tender, 25 to 30 minutes. Scrape insides with a fork to remove spaghettilike strands. Keep warm.
Heat remaining 1 tablespoon oil in a pan over medium heat. Sauté onion until tender, about 6 minutes. Add red pepper and 1 tablespoon each garlic and oregano; sauté 1 minute. Add tomatoes and 1/4 teaspoon each salt and black pepper; bring to a boil. Reduce heat to low; simmer, stirring, until thickened, about 15 minutes. Stir in vinegar.
Increase oven to 450°F; mist a baking sheet with cooking spray. Grind oats in a food processor. Place oats and remaining 1 tablespoon each garlic and oregano and 1/4 teaspoon each salt and black pepper in a bowl. Add beef, egg, and 1/4 cup cheese; stir until just combined. Shape into 12 meatballs and place on baking sheet; bake until lightly browned, about 8 minutes.
Add meatballs to tomato mixture. Cook over low heat until a thermometer inserted in mixture registers 160°F, about 5 minutes. Divide squash among 4 plates; top each with 3 meatballs, sauce, and remaining cheese.