I wouldn't say that that particular property makes it impossible, because superluminal particles can also satisfy that equation. Here's the way I understand it:
Every particle has a given energy E and a given momentum p.
Every type of particle has a rest mass m0. The rest mass is important because for every particle of a given type, the energy and momentum must satisfy the equation (it's called the mass-shell equation):
E^2 - p^2 c^2 = (m0)^2 c^4.
Here c is the speed of light. The velocity of a particle is defined to be v = (p/E) c^2 (that's the momentum divided by the energy, multiplied by the speed of light squared).
With some algebra, you can get from the mass-shell equation:
v^2 = c^2 - (c^6/E^2) m0^2.
Now, E and p are always real numbers. This means that m0^2 is real. There are three possible cases:
- m0^2 > 0. Most types of matter fall into this category (electrons, nuclei, etc.) They are restricted to speeds less than the speed of light.
- m0^2 = 0. Most force carriers fall into this category (gluon, hypothetical graviton, photon). They travel at exactly the speed of light.
- m0^2 < 0. These particles have imaginary "rest mass". They always travel at speeds strictly greater than the speed of light. We don't know of any particles like this.
The experiments that directly measure neutrino mass always give upper bounds (or, put another way, all we know directly about neutrino masses is that they're very small). The experiments that tell us indirectly about neutrino masses don't tell us their masses, but instead the
differences in rest mass squared. Quantum mechanics aside, it is possible for the mass-squared of any of the neutrinos to be negative without having a violation of special relativity alone.
The issue is not with special relativity: it's with quantum mechanics. Quantum field theory says that tachyons should never exist and that they are in fact a sign that the vacuum state you have chosen for it is unstable, and that you need to choose a new vacuum, one which inevitably does not contain any tachyons. Every quantum field theory I am aware of that has "tachyons" is like this.