Place buttery spread and sugars in a large bowl. Using an electric mixer at low speed, cream all ingredients for a minute. Add butternut squash puree and vanilla extract, and mix until everything is just incorporated. This should take no more than 10 to 15 seconds.
In a separate bowl, mix the gluten free flour, baking powder, baking soda, spices, and salt together.
Sift flour mixture into the bowl with the wet ingredients. Mix until all ingredients are just incorporated, and be careful not to overbeat the mixture.
Cover the bowl with plastic wrap and freeze for 30 minutes. Freezing the dough will make it easier for you to scoop and and shape the dough later.
Line two baking sheets with parchment paper. Set aside.
Preheat oven to 350 degrees F (175 degrees C). Position the oven racks to the top third and bottom third slots.
Scoop out a heaping tablespoonful of cookie dough and shape into a ball. Place the dough onto a baking sheet and flatten with your fingers. Repeat for the rest of the cookie dough.
Bake cookies for 10 to 12 minutes or until the edges start to turn golden. I also like to swap the sheets around halfway through so that the cookies get the same amount of heat. Let the cookies sit on the sheet for 5 minutes before cooling them off on a rack.