Let G be a finite classical group, specifically one of the general linear, unitary, symplectic, or orthogonal groups defined over a finite field of odd characteristic. For any element g ∈ G, we consider the fiber of the squaring map at g, which is the set of all elements h ∈ G satisfying h2 = g. In this work, we determine the cardinality of these fibers for arbitrary elements g ∈ G. Our analysis enables us to compute the total number of real conjugacy classes in G. The central tool for this is a recent resolution of Brauers Problem 14.