Heat oven to 350 F. Remove half of cookie dough from wrapper; refrigerate remaining dough until needed. On lightly floured surface with rolling pin, roll half of dough into 9-inch square (make edges square with side of ruler). Cut into 9 (3-inch) squares; place on ungreased large cookie sheet.
With sharp knife or fluted pastry wheel, cut each square from each corner almost to center, leaving about 1 inch uncut in center. Place wooden stick on each square with one end in center. Fold alternate corners of square over, pressing in center. Place gumdrop half in center to resemble button. Sprinkle whirligigs with red and/or green sugar. Repeat with remaining half of dough.
Bake 9 to 11 minutes or until golden brown. Cool 1 minute; remove from cookie sheet and place on wire racks. Cool completely, about 15 minutes.