Add the squash, onion, bell pepper, garlic, black beans, tomatoes and broth to a 6-quart slow cooker.
Sprinkle in the chili powder, cumin, paprika and salt and stir until no dry spice pockets remain.
Cover and cook on low until the squash is tender at the center but most cubes still hold their shape, 5 to 5 1/2 hours.
Mash about 1 cup of the squash and beans against the side of the cooker and stir it back in to thicken the chili.
Stir in the lime juice and cook uncovered on high for 10 minutes if the chili looks loose.
Taste for salt and serve while the squash still has defined edges.