Preheat oven to 375 degrees F (190 degrees C). Grease 12 muffin cups or line with paper liners.
Whisk flour, baking powder, and baking soda together in a bowl.
Place butternut squash in a saucepan and cover with water; bring to a boil. Reduce heat to medium-low and simmer until tender, 15 to 20 minutes. Drain and transfer squash to a bowl; mash with a potato masher until smooth. Add ginger, lemon zest, and lemon juice; mix well.
Mix cream, sugar, oats, and eggs together in a large bowl. Whisk squash into cream mixture until light and creamy. Stir squash mixture into flour mixture until batter is just combined; fill muffin cups 3/4-full.
Bake in the preheated oven until a toothpick inserted in the center of a muffin comes out clean, about 20 minutes. Allow muffins to cool for 5 minutes in the muffin cups before transferring to a serving plate.