Preheat oven to 350°F. Line a muffin tin with 12 cupcake liners.
Combine flours, oats, baking powder, baking soda, pumpkin pie spice and salt in a large bowl, stirring with a whisk. Make a well in the center of the dry ingredients.
Whisk together puréed squash, sugars, oil and eggs in a medium bowl. Add liquid ingredients to dry ingredients, stirring with a spoon, just until combined. (Don’t over mix, or your muffins will turn out tough.) Gently fold in nuts and cranberries. Spoon batter evenly into muffin cups.
Bake for 25 minutes, or until a wooden pick inserted in the center comes out clean. Cool 5 minutes in muffin tin, and then turn muffins out, and cool completely on a wire rack.