Preheat oven to 325 degrees F.
In a bowl, combine sugar cookie mix, melted butter, egg, cream cheese, and flour; mix together to form dough. Separate dough into 2 equal portions and place in 2 different bowls. Add red food coloring gradually to 1 bowl of dough, kneading together until desired shade of red is created. To second bowl of uncolored dough, add peppermint extract and knead together. On a floured work surface, shape each dough into balls and then roll each ball into 1/4-inch-wide ropes, each about 6 inches long. For each cookie, carefully twist some of the red and white ropes of dough together and shape into a candy cane. Spread candy canes out on cookie sheets and bake on the top shelf of the oven for about 10 to 12 minutes. Transfer to a rack to cool before serving.