If $\mid \lambda_i\mid=1$ and $\mu_i^2=\lambda_i$, then $\mid \mu_i\mid=1$? If $|\lambda_i|=1$ and $\mu_i^2=\lambda_i$, then $|\mu_i|=1$?
$|\mu_i|=|\sqrt\lambda_i|=\sqrt |\lambda_i|=1$. Is that possible?

Yes, that is correct. Or, either you could write $1=|\lambda_i|=|{\mu_i}^2|=|\mu_i|^2$, and use $|\mu_i|\ge 0$ to arrive to the unique solution $|\mu_i|=1$.
