I was wondering whether the breakdown field strength for the Meissner effect may be attributed to the Zeeman effect? I can see the latter (along with the Stark effect) to be more analogous to electron screening but would the effect on the density of states due to the reduction of degeneracy have any correlation to the critical field strength that puts a type II superconductor into the phase with quantized vortices?
I think not; the Zeeman effect plays no role here, as that only splits energy levels of atoms. But here (in Ginzburg-Landau theory of Meissner effect) we only have phase transitions and surface currents being produced. In particular, superconductivity fits naturally into a gauge theory, independent of an atom's energy splitting.
The vortex phase can be understood in two different ways (of course strictly equivalent): energy of the surface (does the superconductor (SC) screen the magnetic field) vs. energy of the bulk (is the superconducting phase robust) or London penetration length vs. Ginzburg-Landau coherence length. The vortex pierce the SC when the surface energy becomes smaller than the volume energy (in short, the surface becomes permeable to magnetic field). In the length scale picture, the vortex start to enter the game when the London penetration length becomes larger than the coherence length. Then, the Cooper pairs -- having a coherence length size -- have to adapt themselves to the new situation with part of the magnetic energy in the bulk. So they start to form vortex lattice, since it is energetically favorable for the SC phase.
Both the London and coherence lengths can indeed be Zeeman effect dependent, but here the main problem: the Zeeman effect is really small in comparison with the so-called orbital effect (responsible for the Lorentz force if you wish). A way to kill the orbital effect is to make quasi-2D system, and to apply the external magnetic field in the plane of the system. An other way is to use the so-called heavy-fermions compounds. Both systems kill the orbital effect: the first one because the Lorentz force is perpendicular to the magnetic field and thus directed outside the 2D system, the second-one because the electrons are so-heavy that the magnetic field has difficulties to move them. In contrary, the Zeeman effect is orientation and mass independent in simple models.
So to conclude, you can modify the Meißner effect when Zeeman effect starts to be important, but you first have to find a good situation to ignore the orbital effect.
If I remember correctly, in quasi-2D systems, the Meißner effect is quenched by a strong Zeeman effect partly for the reason you invoked (splitting of the electron level). NB: the Zeeman effect is usually called the paramagnetic effect in condensed matter studies. There is nevertheless still some debates about that. It seems to depend on a lot of effects (impurities, geometry of the compound, band structure, symmetry of the order parameter, ...) I thus prefer to not give you some references.
PS: This answer may complete what Chris Gerig said. I may say for short: when the Zeeman effect enters the scene of superconductivity, everything is a total mess ! But that's make all physicists happy too: how to clean this messy room is a challenge then :-) !