Sprinkle chicken with salt and ground black pepper. Wrap in aluminum foil and place on a baking sheet.
Place butternut squash pieces cut-side down in a 9x13-inch baking dish. Add about 1/2-inch water to the dish and cover with aluminum foil.
Bake chicken and squash in the preheated oven until butternut squash is tender and chicken breasts are no longer pink in the center, about 30 minutes. An instant-read thermometer inserted into the center of the chicken should read at least 165 degrees F (74 degrees C).
Heat oil in a skillet over medium heat. Add onion and garlic; cook and stir until softened, about 5 minutes.
Scoop cooked squash flesh into a large pot. Pour in chicken broth and mash together until smooth using an immersion blender or potato masher.
Cube cooked chicken and add to the pot. Stir in onion, garlic, canned tomatoes, quinoa, oregano, and 1/2 teaspoon salt. Bring soup to a gentle boil; reduce heat and simmer until quinoa is tender and flavors are well-blended, about 1 hour.