I think there is a global symmetry breaking in the BCS theory also. Greiter does says that the BCS has a global symmetry breaking, and http://arxiv.org/abs/0704.3703 by van Wezel and van den Brink says that there is a nearly degenerate thin spectrum that reflects global symmetry breaking in finite superconductors.In the case of the Bardeen-Cooper-Shriefer theory of superconductivity you don't have spontaneous symmetry breaking, because the "broken symmetry" is em. gauge theory.
I think the theory you are thinking about is the Abelian Higgs model which is related to the description of the Meissner effect by Anderson in superconductors, eg. http://arxiv.org/abs/cond-mat/0404327 by Hansson, Oganesyan and Sondhi says that in 3+1 dimensions the Abelian Higgs model "is a plausible description of a gapped BCS superconductor with particle-hole symmetry but it has the topological features of interest even if the choice of a Lorentz invariant dynamics is non-generic."
But what is the ground state degeneracy in the Abelian Higgs model? If I read Hansson et al correctly, they say that the 2+1D Abelian Higgs ground state degeneracy depends on the boundary conditions or the topology of the manifold. The Scholarpedia Higgs article by Kibble http://www.scholarpedia.org/article/Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism does agree that it is misleading but conventional to talk about "spontaneous gauge symmetry breaking" when he refers to an Abelian model, and that it is better to say that there is an explicit breaking of the gauge symmetry by some gauge choices, but the state is gauge invariant. However, he also does say "the resulting theory does retain a global phase symmetry that is broken spontaneously by the choice of the phase of ##\langle\Phi\rangle##."