In large bowl, with electric mixer on medium, beat butter, cream cheese, sugar, egg, and vanilla for 3 minutes. Add flour and continue beating 2 to 3 minutes, until blended. Divide dough into 3 portions.
Wrap 1/3 of the dough and refrigerate. Beat red food coloring into 1/3 of dough; wrap. Clean bowl and beat green coloring into remaining 1/3 of dough; wrap. Chill at least 2 hours, or overnight, until firm.
Heat oven to 375°F. On lightly floured surface, separately roll out the 3 colored doughs to 1/4-inch thickness. Cut out cookies with floured cookie cutters, and place 1 inch apart on ungreased baking sheets. Use scraps of one colored dough to decorate a cookie of another color.
Bake 7 to 10 minutes, until edges are very lightly browned. Remove from baking sheets to wire racks and cool completely.