Negative temperatures are actually a pretty well-defined and real thing, but that's just because of the way we define "temperature" in thermodynamics, which is not always exactly the same as what we think of as temperature in everyday life. The short explanation is that temperature (T) is the rate of change of energy (E) with respect to entropy (S) (in math: T=dE/dS). If I have a system that is bounded from above in energy (i.e. a maximum energy the system can reach), I can get negative temperatures. Simple example: let's say I have a system of particles, each of which can be in two states, a state with more energy, and a state with less energy. The entropy is the number of different states the *entire* system can be in, so if the system is in a minimum energy state (i.e. every particle is in the lower energy state) I have a minimum entropy system (every particle in the same state means I only have 1 possible state for the entire system). Likewise, in a *maximum* energy state, all the particles are also in the same state (the higher energy state), so I also have minimum entropy. Maximum entropy occurs when the energy is right in the middle between these: half the particles are in the higher energy state, half are in the lower energy state, so the entire system has the most possible configurations. So, if the system is in that state, and I add a bit of energy to it, I decrease the entropy (as there are fewer particles in the lower energy state and therefore fewer possible configurations). That means dE/dS is negative (since S goes down, so dS is negative, while dE is positive), so you get negative temperature.
In every day life, systems typically aren't bound from above, and also any particles in higher energy states like that will fall into lower energy states and release energy (this is exactly how a laser works, incidentally), so you only get negative temperature in carefully constructed systems.
The negative mass term in this case, however, is a negative effective mass (not a real mass) term that occurs in a group velocity (which is not the real velocity of particles in the system) dispersion relationship. Not to say the results aren't interesting: they are, they're just... well, not really negative mass at all.