Preheat oven to 350°F. Grease and flour 8- by 8-inch baking pan or line with parchment paper. Sift together flour, baking soda, and spices, and set aside. In separate bowl, mix squash, eggs, brown sugar and applesauce until well incorporated.
Stir flour mixture into wet ingredients until incorporated, then pour into prepared pan.
Bake 20 to 25 minutes, until toothpick inserted in center comes out clean.
Beat together cream cheese, butter, powdered sugar and lemon juice until smooth. Spread on cooled cake.