Preheat oven to 400°F (200°C). Slice squash lengthwise, scoop seeds, brush cut sides with olive oil, and roast cut-side down 35–40 minutes until tender.
While squash roasts, mix chicken, ricotta, spinach, garlic, oregano, mozzarella, Parmesan, salt, and pepper in a bowl.
When squash is cool enough, use a fork to shred strands and leave shells intact. Stir squash strands into filling or stuff shells directly.
Return filled shells to oven and bake 10–12 minutes until top is golden and bubbly. Sprinkle red pepper flakes if using.