Answer: The statement is false.
The box is a rectangular prism with dimensions 2 cm x 2 cm x 2 cm, so it has a volume of 8 cubic centimeters. The statement suggests filling the box with 2 unit cubes in length, 2 unit cubes in width, and 3 unit cubes in height, which would require a total of 2 x 2 x 3 = 12 unit cubes, each with a volume of 1 cubic centimeter. However, 12 cubic centimeters is greater than the volume of the box, which is only 8 cubic centimeters.
Therefore, the statement is false.