Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add butternut squash, cover, and steam until tender, about 25 minutes. Mash squash with a potato masher in a bowl and cool to room temperature.
Preheat oven to 350 degrees F (175 degrees C). Grease 15 muffin cups or line with paper muffin liners.
Beat butter and sugar together in a bowl with an electric mixer until fluffy, 3 to 5 minutes. Beat in eggs, 1 at a time, until mixture is foamy. Pour milk slowly into butter mixture while beating.
Fold mashed squash, baking soda, and nutmeg into butter mixture. Fold flour into squash mixture, folding from the bottom of the bowl to the top. Pour batter into prepared muffin cups.
Bake muffins in the preheated oven until a toothpick inserted in the center of a muffin comes out clean, 18 to 20 minutes.