Preheat the oven to 450 degrees Fahrenheit. Lightly coat a 10-inch cast iron skillet with the vegetable oil and place it in the oven while it preheats. In a separate skillet over medium-high heat, cook the breakfast sausage, breaking it into small crumbles, until browned and fully cooked; transfer to a plate and set aside.

In a large mixing bowl, whisk together the two packages of cornbread mix and the buttermilk, following the package directions but replacing the liquid with buttermilk in the same amount. The batter should be thick and spoonable.

Fold the diced squash, celery, bell pepper, and onion into the cornbread batter until just combined, with the batter lightly binding the vegetables.

Pour the batter into the preheated skillet and spread it evenly. Bake at 450 degrees Fahrenheit for the time indicated on the cornbread mix package, adding a few extra minutes if necessary, until the cornbread is lightly browned and a toothpick inserted in the center comes out clean.

Remove the cornbread from the oven and let it cool in the skillet on a wire rack until just warm. Reduce the oven temperature to 350 degrees Fahrenheit.

Crumble the still-warm cornbread into a large bowl. Add the butter in small pieces and mix until the butter melts and the cornbread is evenly crumbly.

Add the cooked sausage, cream of chicken soup, chicken broth, eggs, poultry seasoning, and black pepper to the cornbread mixture. Stir thoroughly until everything is well incorporated and the mixture resembles a thick batter.

Spoon the dressing into a lightly buttered 9x13-inch casserole dish, spreading it into an even layer. Bake at 350 degrees Fahrenheit for 30 to 40 minutes, or until the top is golden brown and the center is set.