Place Great Northern beans into a large container and cover with several inches of cool water; let stand 8 hours to overnight. Drain.
Heat oil in a large pot over medium heat; cook and stir onion and garlic in the hot oil until onion is translucent, 5 to 10 minutes. Add bay leaves and cook until fragrant, about 1 minute.
Mix beans, water, and bouillon into onion mixture; bring to a boil. Reduce heat, cover pot, and simmer, stirring occasionally, until beans are tender, about 1 hour.
Stir squash, carrots, celery, salt, black pepper, white pepper, ginger, and allspice into broth mixture; simmer until squash and carrots are tender, about 20 minutes.
Remove bay leaves from broth mixture and discard. Scoop 2 to 3 cups of vegetable-bean mixture using a slotted spoon from broth; blend broth and remaining vegetable-bean mixture using a hand blender until smooth. Return the vegetable-bean mixture to broth mixture and stir in sausage. Simmer soup until sausage is cooked through, 5 to 10 more minutes.
