  We use equivariant localization and divided difference operators to determine
formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit
closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one
of the symmetric subgroups $O(n,\C)$ or $Sp(n,\C)$. We realize these orbit
closures as universal degeneracy loci for a vector bundle over a variety
equipped with a single flag of subbundles and a nondegenerate symmetric or
skew-symmetric bilinear form taking values in the trivial bundle. We describe
how our equivariant formulas can be interpreted as giving formulas for the
classes of such loci in terms of the Chern classes of the various bundles.
