Preheat oven to 400°F. In the microwave cook one whole squash 3 minutes. Repeat with second squash. Cut each in half and scrape out seeds. Drizzle 1 teaspoon oil over each half and rub around until insides are coated. Sprinkle each with salt and pepper to taste; place halves, scraped sides down, on a parchment-lined baking sheet center so they lie flat. Bake 50 minutes.
Heat large sauté pan over medium-high. Add 2 teaspoons oil to pan, then chicken, ¾ cup onions, and garlic. Cook until chicken is cooked through.
Add tomatoes, cumin, chili powder, and 1 teaspoon salt; stir to combine. Bring mixture to boiling and cook, uncovered, 15 minutes to reduce liquid, stirring occasionally.
Remove from heat. Stir in sliced olives and lime juice.
When squash is cooked, scrape insides with a fork to release from edge and create a nest. Put ¼ of the chicken mixture and ¼ cup cheese in each squash section.
Bake 5 minutes or until cheese is melted. Top with avocado, diced tomatoes, remaining onions, jalapeño, whole olives, and lime wedges.