Assume that the roots $\zeta$ and $\xi^2/\zeta$ of your equation are roots of unity. Then by Vieta Theorem $\zeta+\xi^2/\zeta=-(\xi^2-\xi+1)$; equivalently sum of 5 roots of unity $\zeta+\xi^2/\zeta+\xi^2-\xi+1$ vanishes. All such relations are classified in Theorem 6 of [paper by Conway and Jones][1]. This implies the result.


  [1]: http://matwbn.icm.edu.pl/ksiazki/aa/aa30/aa3033.pdf