Preheat Oven: Preheat your oven to 375°F (190°C).
Prepare Squirrel: Pat the squirrel pieces dry with paper towels. This helps the coating adhere and ensures better browning.
Prepare Coating:
In one shallow bowl, crack and beat the eggs with a fork until well combined.
In the other shallow bowl, combine the seasoned coating mix, flour, and any optional spices (paprika, garlic powder, black pepper, cayenne pepper, if using).
Brown the Squirrel:
In a large skillet (preferably oven-safe) over medium heat, combine the olive oil, canola oil (or vegetable oil), and butter. Heat until the butter is melted and the oil is shimmering.
Dip each piece of squirrel into the beaten egg mixture, allowing any excess to drip off.
Immediately dredge the egg-coated squirrel in the flour mixture, pressing gently to ensure it's evenly coated.
Place the coated squirrel pieces in the hot skillet, being careful not to overcrowd. Brown the meat on all sides until golden brown, about 15-20 minutes. You may need to do this in batches.
Bake the Squirrel:
Transfer the browned squirrel pieces to a 9x13 inch baking dish.
Make the Gravy:
If using an oven-safe skillet, you can deglaze it directly. If not, transfer the skillet drippings to a regular skillet on the stovetop.
Pour the chicken broth, white wine, or water into the skillet used for browning (or the skillet with drippings).
Bring to a simmer over medium heat, scraping up any browned bits (fond) from the bottom of the skillet with a wooden spoon. This adds flavor to the gravy.
Pour the liquid and any loosened browned bits evenly over the squirrel pieces in the baking dish.
Bake: Cover the baking dish with a lid or aluminum foil. Bake in the preheated oven for 1 hour, or until the squirrel meat is tender and cooked through. Check for doneness by inserting a fork into the thickest part of a piece; the meat should be easily pierced.
Serve: Remove from the oven and serve hot.