jmc

Note that $EJCI$ is a rhombus by symmetry. Let the side length of the cube be $s$. By the Pythagorean theorem, $EC=s\sqrt{3}$ and $JI=s\sqrt{2}$. Since the area of a rhombus is half the product of its diagonals, the area of the cross section is $\frac{s^2\sqrt{6}}{2}$. This gives $R=\frac{\sqrt{6}}{2}$. Thus, $R^2=\boxed{\frac{3}{2}}$.