jmc

First, consider the case where $z$ is between $z^2$ and $z^3.$ The diagram may look like the following:



The arrows in the diagram correspond to the complex numbers $z^3-z$ and $z^2-z,$ which are at $90^{\circ }$ angle to each other. Thus, we can obtain one complex number by multiplying the other by $i.$ Here, $z^3-z=i(z^2-z).$

Another possible diagram is as follows:



Here, $z^3-z=-i(z^2-z).$ Thus, we can combine both equations as $$z^3-z=\pm i(z^2-z).$$We can factor as $$z(z-1)(z+1)=\pm iz(z-1).$$Since the square is nondegenerate, $z\ne 0$ and $z\ne 1.$ We can then safely divide both sides by $z(z-1),$ to get $$z+1=\pm i.$$For $z=-1+i,$ the area of the square is $$|z^2-z{|}^2=|z{|}^2|z-1{|}^2=|-1+i{|}^2|-2+i{|}^2=10.$$For $z=-1-i,$ the area of the square is $$|z^2-z{|}^2=|z{|}^2|z-1{|}^2=|-1-i{|}^2|-2-i{|}^2=10.$$Another case is where $z^2$ is between $z$ and $z^3.$



This gives us the equation $$z^3-z^2=\pm i(z-z^2).$$We can factor as $$z^2(z-1)=\pm iz(z-1).$$Then $z=\pm i.$

For $z=i,$ the area of the square is $$|z^2-z{|}^2=|z{|}^2|z-1{|}^2=|i{|}^2|i-1{|}^2=2.$$For $z=-i$, the area of the square is $$|z^2-z{|}^2=|z{|}^2|z-1{|}^2=|-i{|}^2|-i-1{|}^2=2.$$The final case is where $z^3$ is between $z$ and $z^2.$



This gives us the equation $$z^3-z^2=\pm i(z^3-z).$$We can factor as $$z^2(z-1)=\pm iz(z-1)(z+1).$$Then $z=\pm i(z+1).$ Solving $z=i(z+1),$ we find $z=\frac{-1+i}{2}.$ Then the area of the square is $$|z^3-z^2{|}^2=|z{|}^4|z-1{|}^2={\left|\frac{-1+i}{2}\right|}^4{\left|\frac{-3+i}{2}\right|}^2=\frac{1}{4}\cdot \frac{5}{2}=\frac{5}{8}.$$Solving $z=-i(z+1),$ we find $z=\frac{-1-i}{2}.$ Then the area of the square is $$|z^3-z^2{|}^2=|z{|}^4|z-1{|}^2={\left|\frac{-1-i}{2}\right|}^4{\left|\frac{-3-i}{2}\right|}^2=\frac{1}{4}\cdot \frac{5}{2}=\frac{5}{8}.$$Therefore, the possible areas of the square are $\boxed{\frac{5}{8},2,10}.$