Peel and seed squash. Cut into 1-inch chunks; you should have about 3 cups. In a 3- to 4-quart pan over high heat, combine 1/2 cup water with brown sugar and the squash. Bring to a boil, cover, reduce heat to medium, and cook, stirring once, until squash is almost tender when pierced, 8 to 10 minutes. Uncover and cook, occasionally stirring gently, until most of the water is evaporated, 3 to 5 minutes longer. Remove from heat and let cool.
Peel and chop onion. Rinse, stem, seed, and chop bell pepper. Rinse fennel and trim off and discard base of head and tough stalks. Chop enough fennel to make about 1 cup. Rinse mushrooms and slice 1/4 inch thick.
Squeeze sausage from casings and break into chunks into a 12-inch frying pan or 5- to 6-quart pan over medium-high heat; stir often until lightly browned and crumbly, 7 to 10 minutes. Pour sausage into a wire strainer to drain; discard fat.
Preheat oven to 350°. Add butter to unwashed pan over medium-high heat; when melted, add onion, bell pepper, and fennel and stir often until vegetables begin to soften, 7 to 8 minutes. Add mushrooms and stir until limp, about 3 minutes. Remove from heat and stir in sausage, cilantro, sage, and thyme. Add squash.
In a large bowl, whisk together eggs and broth. Break cornbread into 1/2-inch chunks and drop into bowl. Add sausage mixture and stir gently to combine. Scrape dressing into a buttered 9- by 13-inch baking dish and cover with foil.
Bake until hot in the center, 40 to 50 minutes. Uncover and bake until top is crisp and golden brown, 10 to 15 minutes longer.
Cornbread: Preheat oven to 400°. In a bowl, mix 2 boxes (8 1/2 oz. each) corn muffin mix, 3/4 cup milk, and 2 large eggs until blended but still slightly lumpy. Pour batter into a buttered 9- by 13-inch baking pan. Bake until top is golden brown, 15 to 20 minutes. Let cornbread cool in pan on a rack.