Fibers of the square map in some finite groups of Lie type and an application

Let $G$ be one of the finite general linear, unitary, symplectic or orthogonal groups over finite fields of odd order. We find the cardinality of the fibers of the square map at a given generic element. Using this we find the number of real conjugacy classes of $G$. This is primarily achieved by leveraging a recent solution to Brauer’s problem 14.