I think in this thread everything is very confused. The original question was, how Maxwell came to the prediction of em. waves and why it let him to conjecture that light might be such an electromagnetic wave. Then it drifted away to a discussion about special relativity, because perhaps, the question has been posted in the relativity subforum.
Of course, Maxwell had no idea about relativity. He just used the collected empirical wisdom (particularly the very detailed experimental results and visionary qualitative theoretical ideas leading to the field description by Faraday) and also previous mathematical models about electricity and magnetism and wrote down a new dynamical model of these empirical facts, based on very complicated mechanical models involving a socalled "aether", i.e., a substance with very "exotic" properties.
The equations were in essence of course what we now call Maxwell's equations, but they were written down in this form only later by Heaviside, who also introduced modern vector calculus. The fascinating thing about Maxwell's theoretical prediction of em. waves is that it rested entirely on static empirical input and the most important addition to the previous models (Ampere, Neumann), i.e., the famous "displacement current". This lead, of course using the then used electrostatic and magnetostatic units, Maxwell to deduce the existence of electromagnetic waves with a phase velocity that was equal within the then established accuracy to the speed of light in vacuum (or rather air, which is not too different, particularly not in view of the then established precision of measurement). The numerical value of the speed was deduced from the comparison of the measures for electric charge in electrostatic and magnetostatic units. The most accurate experiment was by Weber and Kohlrausch (1855). In the language of the modern SI, what was established is the relation ##c=1/\sqrt{\mu_0 \epsilon_0}##, where ##\mu_0## and ##\epsilon_0## were measured at this time!
Another question is about Einstein's arguments leading him to the two famous postulates of 1905 ("On electrodynamics of moving bodies"). Einstein was very clear about the motivation resting on symmetry arguments (which brought into physics a line of thought that was of utmost importance for the entire modern physics, including quantum theory, based on Noether's famous theorems, which were inspired by the complicated question about the energy of the gravitational field in connection with Einstein's General Theory of Relativity). The problem was the lack of Galilei invariance of Maxwell's equations. Now, Einstein's idea was that the special principle of relativity should still be valid. If you now look at Maxwell's equations in their more lucid form when writing them in Gaussian or Heaviside-Lorentz units (which was the usually used units around 1900), you see that then the speed of light, which occurs as a natural constant in Maxwell's equations, must be invariant, implying that the speed of light cannot depend on the speed of (at least uniformly) moving light sources. From these two postulates Einstein could deduce the Lorentz transformation between inertial reference frames. Einstein bluntly gave up the very foundation of (Newtonian) physics, namely the spacetime model of Galilei and Newton. There was necessarily no absolute time and space anymore to make the special principle of relativity compatible with the "constancy of the speed of light". This implied that all of physics, particularly mechanics, had to be changed rather than Maxwell's equations, which were known to be invariant under Lorentz transformations even before Einstein (it was found up to a detail already by Woldemar Voigt around 1900).