In a 5-quart stockpot, soak the dried beans in water, covering by 2 inches. Soak for 8-12 hours. After the beans are soaked, drain them.
To the beans, add chicken stock. Bring to a boil, then reduce to a simmer. Cover and cook until the beans are tender and creamy, about 30-45 minutes.
While the beans are cooking, in an 8-quart stockpot, heat oil over medium heat. Add onion, carrots, and salt. Sauté until softened and translucent, 3-5 minutes.
Add cumin, Mexican oregano, and chili powder. Stir until fragrant, about 30 seconds.
Add chopped pepper, tomato purée, and squash. If the mixture seems thick, add ½ cup water. Bring it to a simmer.
Add the chicken breasts by nestling them into the soup. Cover and cook.
While the chicken is cooking, check the beans. When they're tender, add the beans and their cooking liquid to the chili with the chicken. Stir and continue to cook until the squash is easily pierced with a knife, and the chicken registers 160°F.
When the chicken is done, remove it from the chili, and shred it with two forks. Add the shredded chicken back to the chili.
Serve the chili hot with sliced avocado, cilantro, sour cream, or anything else desired.