The choice of the values for \epsilon_0 and \mu_0 is some men made convention to get convenient quantities for currents, voltages, and charges in electrical engineering, something Nature couldn't care less about.
The only fundamental constant that enters the theory of electromagnetism compared to Newtonian mechanics is the universal speed of light in the vacuum. As it has come out through Einstein's SRT paper of 1905, this is a much more universal speed than just the speed of light in vacuum.
More recent analyses of the special relativity principle has shown that there are, up to equivalence, only two space-time manifolds that obey the special relativity principle (i.e., the indistinguishablitiy of all inertial frames, i.e., the impossibility to measure an absolute constant velocity) are either Galilei-Newton space-time (a fibre bundle, i.e., you simply pin 3D Euclidean spaces at each point along the time axis) or Einstein-Minkowski space-time (a pseudo-Euclidean affine manifold with a fundamental form of signature (1,3) or equivivalently (3,1), depending on which sign convention you prefer). The latter implies a universal speed c, which to the best of our knowledge is the phase velocity of electromagnetic waves in the vacuum. This is because empirically there is no hint for a non-zero photon mass (to put it in modern QFT languague).
This confusion can be avoided by using the more natural Gaussian or the rationalized Gaussian (Heaviside-Lorentz) system of units, where no artificial constants due to the choice of an arbitrary fourth independent unit for electric current (in the SI the unit Ampere) or charge (in the SI the unit Coulomb=Ampere times second) is introduced. That's why not long ago usually textbooks on theoretical classical electrodynamics were written using the Gaussian units and why in theoretical high-energy physics one usually uses the Heaviside-Lorentz system of units. Nowadays, unfortunately the textbooks on classical electromagnetism are written using the SI units, spoiling somewhat the beauty of Maxwell theory as a relativistic classical field theory.
In my opinion, the best compromise is found in the third edition of Jackson's textbook: He starts using the SI units in order to be compatible with the use of this units in experimental physics and engineering and then switching to Gaussian units in the chapters where the electrodynamics is treated in its true form as a relativistically covariant field theory, writing "SI" or "G" in the header line of the chapters :-).
