Could a non-photon massless particle travel at a speed other than $c$? The speed of light is given as $c=\frac{1}{\sqrt{ε_0μ_0}}$ which is in terms of the electric and magnetic constants.
Hypothetically, another massless particle could exist which does not interact with the electronic or magnetic fields. Postulate that a "foo" field and a "bar" field exist which were mathematically analogous to the electric and magnetic fields, the "foobaron" particle might have a speed given by $c_{fb}=\frac{1}{\sqrt{f_0b_0}}$, derived in exactly the same way that $c$ is derived, but using different, independent fields.
If such a particle were to exist, must $c=c_{fb}$? Why?
 A: In special relativity, the relativistic factor
$$
\gamma = \frac1{\sqrt{1-v^2/c^2}} = \frac{E_\text{total}}{mc^2}
$$
is just as closely related to the particle’s total energy as to the particle’s speed. Any object whose total energy is very much larger than its rest mass will be traveling near $c$. Objects whose rest mass is identically zero are a special case of this limit.
The speed limit $c$ is not a property of electromagnetism; it is a property of spacetime. If a massless particle were traveling at less than $c$, I could hop into my massive rocket ship and catch up to it.  In its rest frame, it would not be a massless particle.  A contradiction, because mass is invariant under Lorentz boosts.
A: c is the speed of cause and effect. See Do we know why there is a speed limit in our universe?
The speed of light is a special case of this. If you move a point charge, the "news" that it has moved will spread out at this speed. That is, changes to the electric and magnetic fields from the charge will spread out at this speed. Forces from the charge acting on another charge will not change until the "news" arrives. See In what medium are non-mechanical waves a disturbance? The aether? for more.
It is possible for waves and such to travel at slower speeds. For example sound does. This is because it is a pressure wave. Atoms bump into other atoms and push them. It takes a while for atoms to accelerate. The pushing arises from electric fields in the atoms' electrons. The rate at which neighboring atoms begin to feel the push spreads out at the speed of light.
