Combine sugar and juice drink in a medium saucepan; bring to a boil over medium heat, stirring until sugar dissolves. Add cranberries and next 6 ingredients; return to a boil. Reduce heat, and simmer, uncovered, 5 minutes or until thickened, stirring often. Cool completely (mixture will thicken as it cools).
Place 1 sheet phyllo on a damp towel; brush with butter. Layer a second and third sheet on top of the first, brushing each with butter. (Set remaining 3 sheets aside, covering with a damp towel to prevent drying out.)
Cut phyllo in half lengthwise; cut each half crosswise into thirds, forming 6 squares. Place 2 pieces of Brie in center of each square. Gather corners of phyllo squares over Brie, and twist gently, forming a bundle. Place bundles on a lightly greased baking sheet. Repeat procedure with remaining phyllo, butter, and Brie.
Bake at 375° for 13 minutes or until golden. Place 2 bundles on each of 6 individual serving plates, and serve hot with cranberry chutney.