Put the beans into a bowl and add enough water to cover by 2 inches. Soak for 8 hours, then drain. (For a quick soak, bring the beans and water to a boil over high heat and cook for 2 minutes, then remove from heat and allow to stand for 1 hour. Drain.)

Heat the oil in a large heavy pot or Dutch oven over medium-high heat. In batches, cook the chicken, seasoning with the salt and pepper and turning once, until evenly browned, about 10 minutes. Transfer the chicken to a platter. Pour off all but 2 tablespoons of the fat from the pot.

Add the onions, carrots, and celery. Cook, stirring often, until softened, about 3 minutes. Add the garlic and stir until fragrant, about 1 minute. Stir in the zucchini and yellow squash and cook for 1 minute. Add the stock, drained beans, tomatoes, parsley, bay leaves, and cayenne. Return the chicken to the pot. Bring to a boil over high heat. Reduce the heat to medium-low and simmer, uncovered, until the beans are almost tender, about 1 hour.

Using a long-handled slotted spoon or tongs, carefully transfer the chicken pieces to a plate. Cool slightly, then remove the meat from the bones, discarding the skin and bones.

Return the chicken meat and any accumulated juices to the pot. Add the rice, green onions, thyme, parsley, and spinach and simmer until the rice and beans are tender, about 15 minutes. Discard the bay leaves.

Ladle into warmed bowls and serve hot.