First we know that surface area is 6 times the area of each face, or $ 6s^2 $, and we set this equal to $ \frac{1}{6} $ of the volume. $$6s^2=\frac{1}{6}s^3\qquad\Rightarrow 36s^2=s^3 \qquad\Rightarrow s=36$$Now we want a square of side length $ a $ and area $ a^2 $ to have an area equal to the volume of the cube. $$a^2=s^3=36^3=(6^2)^3=6^6\qquad\Rightarrow a=\sqrt{6^6}=6^3=216$$So the side length of the square should be $ \boxed{216} $.