Strong Force constants which derive to the speed of light?

This is always a good question. Keep in mind that μ₀ and ε₀ are experimentally determined constants, and their involvement in the determination of the speed of light is a direct result of the wave solution of Maxwell’s equations. The value of ‘c’ then is an experimentally determined value which leaves it open as to why that particular value appears to be the speed limit of causal influences.

So, it begs the question of why all massless particles travel at the speed of light - and they all did prior to the symmetry breaking of the Higgs field. Why that value and not some other value?

Along the lines of Dr Zorro’s argument, light takes the null path in spacetime (ds² = 0) and we conclude that all massless particles must take that null path, putting an upper limit on the speed of causality. But, there is no derivation for each type of massless particle involving constants (such as  μ₀ and ε₀ ) as we have for light.

Very interesting question. The strong force must have a strong force wave and the quantum particle of that wave is the gluon. The wave must have a wave equation analogous to the electromagnetic wave equation. This wave equation will have a velocity term, which must be C. 

How to get this wave equation and the strong force constants involves Quantum Chromodynamics. This is way beyond my level of understanding. 

Gluons are not particles in same way as photons. They are _fields_. Or perhaps it's better to say that gluon fields give rise to phenomena that we call particles. Single gluons don't last long enough for their existence to qualify as a particle that could have a speed.

Well, I don't know and this is beyond my safe physics level, but anyway...

The Strong Nuclear force only operates over such tiny distances that it really isn't much help to consider some universal constant of some all pervading field, when in reality it doesn't show up at all unless you are inside protons or neutrons.

The fact that it is considered a force indicates there would be a Classical field way to think about it in the general case and presumably this field would have some constant(s) that define a speed-of-light sine wave. But in practice I think that inside atoms we are always going model the Strong force in terms of Quantum Chromodynamics and outside of atoms we just don't see it.

It is from special relativity that we know that all massless particles travel at the same speed c. Because photons are massless particles and from classical electromagnetism we know that this speed is the speed of light, we conclude that also gluons travel at that speed.

The speed of light should better be called the speed of causality, because that is what the cosmic speed limit is.