In a 3-1/2- or 4-quart slow cooker combine squash and mushrooms. Stir in undrained tomatoes, drained Great Northern beans, the water, garlic, Italian seasoning, and pepper.
Cover and cook on low-heat setting for 8 to 10 hours or on high-heat setting for 4 to 5 hours.
Shortly before serving, for dumplings, in a medium bowl stir together flour, cornmeal, cheese, parsley, baking powder, and salt. In a small bowl combine egg, milk, and oil. Stir egg mixture into flour mixture just until combined.
If using low-heat setting, turn to high-heat setting. Stir in frozen green beans. Drop dumpling dough by tablespoons on top of mixture in cooker; sprinkle with paprika. Cover and cook for 50 minutes more, leaving the lid on the entire time.