Why does Special Relativity apply to more than just light? It is my understanding that time dilation is derived from the constancy of the speed of light in vacuum. I would assume this implies that the quirky consequences would therefore apply only to light. But they don't. They apply to all material objects. Why?
 A: The basic idea is that physical laws are same in all inertial frames. Framing your question in a different way: Why do we generalize a formula(which gives time-dilation) whose derivation is based on a light clock to physical clocks and even the biological clock?
A very interesting argument was given by Feynman in his Lectures on Physics, Vol:1.

""To answer the above question, suppose we had two other clocks made exactly alike with wheels and gears, or perhaps based on radioactive decay, or something else. Then we adjust these clocks so they both run in precise synchronism with our first clocks (the light clock). When light goes up and back in the first clocks and announces its arrival with a click, the new models also complete some sort of cycle, which they simultaneously announce by some doubly coincident flash, or bong, or other signal. One of these clocks is taken into the space ship, along with the first kind. Perhaps this clock will not run slower, but will continue to keep the same time as its stationary counterpart, and thus disagree with the other moving clock.
Ah no, if that should happen, the man in the ship could use this mismatch between his two clocks to determine the speed of his ship, which we have been supposing is impossible. We need not know anything about the machinery of the new clock that might cause the effect—we simply know that whatever the reason, it will appear to run slow, just like the first one.""

A: The special theory of relativity is a new theory of the space and time and all phenomena in the spacetime. The key constant that determines how strong the new "quirky" effects are is the speed 
$$ c = 299,792,458\,{\rm m/s} $$
which is the speed by which the "spatial directions" may be converted to the "temporal direction" in the spacetime. Whenever the speed of object is at least comparable to $c$, the new "quirky" effects arise and cannot quite be neglected. The closer $v$ is to $c$, the stronger the effects become. Here, $c$ is the maximum speed that the information may propagate by according to the special theory of relativity. And it's also the speed that may never be surpassed – and not even quite reached – by massive objects. But the massive objects' speed $v$ may approach $c$ arbitrarily closely.
This speed $c$ is normally referred to as the "speed of light in the vacuum" because this was the first way how this speed appeared in physics. 100 years ago, people knew basically nothing else than light that could have moved this fast. 
But that doesn't mean that the constant $c$ has some special relationship with light only. There are many other things that it's the speed of. I've already mentioned it. It is also the "limiting maximum speed for massive objects" – and the constant could be called in this way, too.
Also, it's the speed of the gravitational waves. Much like electromagnetic waves, gravitational waves – for example those whose discovery by LIGO was announced in February 2016 – propagate by the speed $c$. In fact, many physicists would agree that it would be far more conceptually meaningful to use the name "the speed of gravitational waves" for the constant $c$. The speed $c$ is the speed of light in the vacuum but it is not "just" the speed of light in the vacuum. It's the maximum allowed speed of information of any form, the speed of the gravitational waves, the speed of other things that may exist (and 99.9999972% of it is the speed of the protons at the LHC). It's a fundamental conversion factor between space and time, energy and momentum etc. that is important in all of physics. The fact that the light in the vacuum propagates by this limiting speed is really a consequence of the special theory of relativity. It's one important but surely not isolated example of the omnipresence of the constant $c$ in fundamental physics, not a unique meaning of the constant $c$.
Planck's constant $\hbar$ is similarly omnipresent in Nature thanks to the theory known as quantum mechanics. It's the energy-to-frequency ratio of photons and all other particles, it's the quantum of the angular momentum (minimum nonzero amount for a boson), it's the natural unit of the action in the principle of least action, it's the minimum product of the uncertainty of the position and momentum, and many and many other things. All these descriptions may be shown to be governed by the same constant. The constant could be called "the quantum of the action" or "quantum of the angular momentum" etc. but we just use the neutral "Planck's constant". 
Similarly, $c$ could very well be called "Einstein's constant", to remove the unfairly close relationship with light. The only problem would be that this name would be highly ahistorical. The constant $c$ was present in physics and approximately measured centuries before Einstein was born. He was the man who appreciate all the numerous roles that the constant plays in physics but he wasn't the first one who encountered the constant.
