Are gas giants supported by thermal pressure? I've heard gas giants are supported because there is an equilibrium between thermal pressure and gravity. That is, if Jupiter were to suddenly begin compressing, temperatures would increase to the point that it would expand to its original size. Since it is in equilibrium, both the compression and thermal pressure counter one another, so its size never fluctuates.

However, Jupiter's average temperature is extremely low, so much so that its gases are electron degenerate. Is this a contradiction, or has the information above been just hearsay?
I am not sure what you mean by "thermal" pressure. Jupiter is supported by pressure, just like all objects that are in (approximate) hydrostatic equilibrium.

That pressure is provided by your everyday, temperature-dependent Maxwell-Boltzmann ideal gas pressure in the outer parts, but the free electrons in the interior become degenerate and so in these regions I suppose you would more accurately describe the pressure as being due to (partial) electron degeneracy pressure.

At the very centre there may be a liquid or even a solid core. In the metallic hydrogen model then this would still be degenerate electrons contributing the pressure. For a solid, rocky core, well solids are rather incompressible.

You can only describe Jupiter as being in approximate hydrostatic equlibrium. It is losing energy from its surface; this energy is supplied from gravitational potential energy and Jupiter is shrinking at a rate calculated to be about 2 cm per year. As it does so, the interior gas becomes denser and more and more degenerate and the pressure will become more independent of temperature. As a result, the rate of contraction will slow down and Jupiter will tend towards the radius of a "cold" body of that composition.