Why is metallic hydrogen degenerate matter? Why is metallic hydrogen considered a form of degenerate matter, akin to neutronium and electron-degenerate matter? I can understand that for the other two, degeneracy pressure is the only force countering inward gravity for very massive stars, but how does this concept also apply to super pressurised hydrogen? Furthermore, why would being supported by degeneracy pressure make the hydrogen metallic in nature? 
 A: If you crush hydrogen to very high densities, the Pauli exclusion principle will prevent electrons in the material from occupying the same quantum states. The electrons will obey Fermi-Dirac statistics and they will fully occupy states up to the Fermi energy - which increases as the density gets higher.
Presumably in metallic hydrogen what happens is that the Fermi energies increase significantly above any binding energy with the protons and thus the electrons essentially become a degenerate free electron gas. The conductivity of such a gas would be very high since degenerate electrons have very long mean free paths. 
On the other hand if the temperature of the protons is reasonably low then it might be possible to confine them in a lattice governed by Coulomb forces. 
The combination of these two things could be considered "metallic" behaviour. A liquid metallic hydrogen would be one in which the ratio of Coulomb to thermal energy in the protons was not high enough for them to "crystallise".
Papers I have seen talk about a possible transition taking place at hundreds of GPa at 300K. If this were a perfect gas of hydrogen atoms, the electron number density would be $\sim 10^{31}$ m$^{-3}$, with a Fermi kinetic energy of $3\times 10^{-17}$ J (187 eV). The degeneracy parameter, $E_F/kT \sim 7000$, would represent a very degenerate gas.
So the answer to your question is that it is electron degeneracy that can give hydrogen "metallic" properties, but to make the electrons degenerate you need get them to very high number densities and to do this you need to squash the gas.
