Is my argument deducing from Maxwells equations the exclusion of faster than light travel flawed? When Maxwell's equations are solved, one of the solutions is electromagnetic waves that should move at a certain speed ($c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$). Now, one could argue that since Maxwell's equations hold for all observers regardless of their reference frame, they should all see these waves with speed $c$. So, the speed of these waves must be independent of your reference frame. 
Moreover, let's say you could observe someone travelling at more than this speed of electromagnetic waves. Now, someone in this reference frame produced these waves. Also, there is a wall in front of them that is destroyed if the electromagnetic wave touches it (it is a powerful laser). And if the person in this reference frame hits the wall, he dies. Now from your perspective, he will die since he is travelling faster than the wave and will reach it first. However, the person himself would be sure he could survive if he fired the electromagnetic wave at the wall before he got to it. So, he would be sure he would not die. 
This leads to a contradiction and so it must be impossible to observe a reference frame that travels faster than this speed of the electromagnetic waves. 
I suspect there is probably a hole in this line of argument but I can't imagine what it is. 
 A: 
Do Maxwells equations imply the constancy of the speed of light?

Maxwell's equations are a rigorous mathematical system and yes, it can be shown that they can have a Lorentz covariant form, after some mathematical transformations.
Then there are the Lorentz transformations themselves, that were necessary to explain the results of the Michelson Morley  experiment, no luminiferous aether, in 1881.

He or someone else should have reached the conclusion that nothing can travel faster than light shortly after.

It takes some time for physics to shift a paradigm,like the existence of luminiferous aether, which was a basic belief since ancient times,  and it was not before special relativity that things fell into place.
Progress in physics is incremental and dependent on mathematics.
I do  think your gedanken experiment is fine.  At the  period of the discovery of Maxwell's equation, there was no fashion of  using gedanken experiments. The Reductio ad absurdum  was not used in physics, which had recently ( 100 years?)discovered the power of calculus and differential equations. Until the Michelson Morley experiment the ether was a very real basis for the physics at that time,  it was a medium through which everything traveled.
We learned to play with thought experiments  due to special relativity and quantum mechanics. At that time there was not even a dream that light could be destructive, for example, lasers were not in that time  frame of reference. That is what I mean by paradigm shift. 
A: The speed of light of has been reduced to zero.  In your argument, an observer, with any positive speed, firing a monochromatic probe beam of unlucky frequency through this medium to detect the wall will hit the wall before the beam does.  Your argument then concludes that any positive speed for the observer is impossible.
Nothing in Maxwell's equations fixes the speed of light.  As soon as we observe two different speeds of light, we have two different speed limits in your argument.  Which one is the real speed limit?
It is a triviality to observe matter and light traveling at various speeds, including matter apparently exceeding the speed (more precisely, phase velocity) of light in a medium (Cherenkov radiation, linked below).  You write "one could argue that [... Maxwell's equations satisfy special relativity ...]".  You assume Special Relativity at this point in your argument.  Of course, if you assume SR, don't be surprised when you get SR.  As counterpoint, what makes you so sure Maxwell's equations aren't an effective theory, only valid for low intensities, like Newton's gravitation?  or large distances (more than, say, ten wavelengths)?  or where the luminiferous ether is particularly rarefied?
When you write "... they should all see these waves with speed $c$."  I believe you mean $c_{\omega,\theta}(O, \Lambda)$, where $\omega$ tracks the frequency dependence (see dispersion), $\theta$ is the polarization (see birefringence), and $O$ and $\Lambda$ record (at least data on the zeroeth, first, and second derivatives of) the reference frames of the observer and the light interacting with matter (see Cherenkov radiation) relative to the luminiferous ether.  And there may well be additional parameters we need to track, like intensity.  Since your argument becomes (at least) ambiguous as soon as one has observations of different speeds of light, all of these cause trouble for your assumption of SR.
A: To answer your title question, definitely not.
(Note: The title question has since been updated. The original was - "is it possible to deduce the constancy of the speed of light from Maxwell's equations?")
If you assume Maxwell's equations hold for all inertial frames as you are doing in your argument, then you are begging the question. You are making an assumption further to Maxwell's equations. That assumption cannot be deduced from Maxwell's equations themselves - you then need to ask how Maxwell's equations transform between different reference frames and that answer is outside the scope of Maxwell's theory. 
In the 19th century, as hinted by dmckee's comment:

You can derive equations that give you the speed of sound in fluids and solids. These equation also hold for all observers in all reference frames (in the context of Galilean relativity). But such waves don't give rise to the surprises in SR. The 19th century physicists assumed they would deal with light in the same way that they had dealt with all the other wave phenomena they had met up to that time. 

people simply assumed that Maxwell's equations held for an observer who was stationary with respect to the luminiferous aether, and that this aether would behave just as gas or the wood of a violin's sounding board for sound. Maxwell's equations would then change their form under the Galilean transformation. That was just "what waves did" to the mind of a physicist in 1862.
Physicists also assumed that Galileo's postulate, that only relative motion between observers were experimentally detectable, didn't hold for electromagnetism. Galileo, after all, knew nothing of electromagnetism.
And that is a perfectly plausible theory to postulate: from a logic standpoint, it is just as valid as Einstein's second postulate and the restoration of Galileo's relativity principle. These things cannot be settled by logic, but only by asking Nature for her take on these things. That is, they can only be settled by experiment - and we find that Maxwell's equations do indeed transform covariantly and that the transformation between relatively moving inertial frames is the Lorentz, not Galilee, transformation.
A: I think the hole in the argument is the affermation that Maxwell's equations hold true for all reference frames.
AFAIK they don't.
They hold true in what Einstein referred to as a local reality.
