Preheat oven to 400°F. Brush squash halves with olive oil, season with salt and pepper, and roast face down 30–35 minutes until tender.
Use a fork to fluff strands into a bowl; reserve shells. Mix chicken, enchilada sauce, black beans, corn, chili powder, cumin, and half the cheese into the squash strands.
Spoon filling back into squash shells, top with remaining cheese, and bake 8–10 minutes until cheese melts and filling is heated through.
Garnish with cilantro, green onions, and a squeeze of lime before serving.