Is there a phase transition for degenerate matter? Does electron or neutron degenerate matter undergo a phase transition? Are there any thermodynamic quantities that are discontinuous? 
 A: Neutron degenerate matter can undergo a phase transition to a superfluid state. The process is thought to be analogous to Cooper-pairing, but the coupling interaction due to the long-range nuclear force is of order 1 MeV, so can occur at temperatures below about $10^{9}$ K in neutron star interiors. The neutrons (fermions) form bosonic pairs in an analogous way to a He-3 superfluid. The neutrons in the deep interior (which dominate the interior 100:1) can form a superfluid; the protons can form a superconducting superfluid. Neutrons in the inner crust can also form a superfluid that has been implicated in pulsar glitches. 
Superfluidity immediately changes the viscosity and the heat capacity of the degenerate neutrons, since it is only neutrons at the top of the "Fermi sea" that pair up. These pairs then have to be broken again to extract whatever heat remains. Ironically, the phase change, which will occur over some period of years in a neutron star because of radial density and temperature variations, causes a period of rapid cooling due to the neutrinos produced by the formation and thermal breaking of neutron pairs. This has now been observed (or claimed, see papers by Heinke & Ho 2010; Shternin et al. 2011).
In the crust of a neutron star there is a phase change associated with it being energetically feasible for degenerate neutrons to drip out of neutron-rich nuclei to join the degenerate electron fluid. This phase change, at densities of about $4\times 10^{14}$ kg/m$^3$, significantly changes the adiabatic index of the gas through electron capture.
At even higher densities ($>10^{16}$ kg/m$^3$) there are thought to be a series of phase changes, collectively called the nuclear pasta phase. This is where the remaining nuclei and neutron fluid arrange themselves into a weird variety of planes (lasagne) or tubes (spaghetti) in order to minimise the total energy density of the material. I am not aware of any major thermodynamic discontinuities associated with this. Eventually the pasta loses its structure to form a fluid of degenerate nucleons (dominated by neutrons) plus degenerate electrons.
Neutrons near the core of a neutron may make a phase change from a fluid state into a structure that is more like a solid, with a very hard equation of state. The details are tied up in uncertainties in many-body strong nuclear interactions and whether the neutrons will start to decay into hyperons or heavy hadrons.
In white dwarf electron degenerate material there is a very important phase change associated with crystallisation of the non-degenerate ions. This increases the heat capacity, releases latent heat of crystallisation and prolongs white dwarf cooling. This crystallisation process is generally accepted fact and is required to match white dwarf luminosities in clusters of known age and the luminosity function of nearby white dwarfs. It has also been detected by asteroseismology because the phase change occurs at higher densities in the white dwarf core and spreads outwards as the white dwarf cools and causes a discontinuity in the interior sound speed (e.g. Winget et al. 1997; Metcalfe et al. 2005).  However, as far as I know, the electrons do not undergo any phase change. They are obviously much too hot to form the Cooper pairs responsible for super conductivity in laboratory materials. 
A: At least in the context of ultracold atomic fermions, the answer is no. The creation of a degenerate fermi gas is, unlike a BEC, not a phase transition.
One major caveat: if there is an attractive force between the fermions, one can get a BCS-like phase transition to condensation of paired fermions. This is, of course, the case for electrons in metals, as well as fermionic liquid He-3.
