Are the permittivity and permeability of free space associated with the zero point energy field? When Maxwell developed the electromagnetic wave equation, he took the wave velocity to be dictated by the permittivity and permeability of the aether.  But the Michelson-Morley experiment demonstrated that there was no aether. This being the case, are the permittivity and permeability of free space (i.e. vacuum), which determine the speed of electromagnetic waves, dictated by the quantum mechanical zero point energy field?
 A: Vacuum permittivity and permeability are just constants of proportionality to relate electromagnetic units (such as fields) to kinematic units (like forces). In SI, they arise from the definition of the Ampere which is defined such that currents (and voltages) have decent units (in SI) to work with in the day-to-day life. You can always change your unit system (like switching from SI to CGS), this will cause a change in the value of those constants ($\epsilon_0$ becomes $\frac{1}{4\pi}$ and $\mu_0$ becomes $\frac{4\pi}{c^2}$). It is even possible to chose a system of units such that one of them is not necessary (e.g. $\epsilon_0=1$). 
Often in physics, we like to put fundamental quantities (like the speed of light) to 1. This simplifies equations at the cost of changing the units (if $c=1$ mass and energy have the same units).
As you can see, there is absolutely no links between the value of some constants and QFT. Those constants are defined within a units system and thus can change from one system to another. There is even units systems where a bunch of constants can be put to 1. Those systems are called "natural units" and a few examples are given here.
A: No, the vacuum permittivity and permeability have nothing to do with quantum mechanics. These dimensionful parameters only exist because of historical reasons: we chose an arbitrary system of units, and the result is the presence of some redundant parameters in the equations.
A wiser choice of units is the Gaussian system, where neither $\epsilon_0$ nor $\mu_0$ appears in any formula: these constants are reabsorbed into the electric and magnetic fields. The mere fact that one can formulate a complete theory of electric and magnetic phenomena with no reference to $\epsilon_0$ and $\mu_0$ implies that the actual value these constants has no intrinsic meaning.
In gaussian units, Maxwell's equations read
\begin{align}
\nabla\cdot\boldsymbol E&=4\pi\rho\\
\nabla\cdot\boldsymbol B&=0\\
\nabla\times\boldsymbol E&=-\frac{1}{c}\frac{\partial\boldsymbol B}{\partial t}\\
\nabla\times\boldsymbol B&=\frac{4\pi}{c}\boldsymbol J+\frac{1}{c}\frac{\partial\boldsymbol E}{\partial t}
\end{align}
and the Lorentz force is
$$
\boldsymbol F=q\left(\boldsymbol E+\frac{1}{c}\boldsymbol v\times \boldsymbol B\right)
$$
As you can see, these equations encapsulate the whole theory of electrodynamics, and they make no reference to the vacuum permittivity and permeability of the vacuum. These parameters only have historical relevance, not physical.
A: I have seen Maxwell's Equations written in at least 4 different ways.   I'm not sure that excluding the showing explicitly of vacuum permittivity and permeability proves anything about those items.  You would then need to show that you don't need them to construct capacitors and inductors -- thinking of course in simple terms:  C = epsilonA/d  and L = mu(N^2*[4*pi*r^2])/l, N = turns, r = radius, and l = length of a coil.
More interestingly a German scientist named Heim used no more than Planck's Constant, the Gravitational Constant, and epsilon_0 and mu_0 above to derive the masses of the three neutrinos of current solar physics interest.  I plugged them into a basic Cosmology Model (of my own construction).  Result was the predicted value of the radius of a neutron = 1.27E-15 meters.  Generally researchers [first] assume a radius for the neutron (commonly 1.25E-15 m) and find a Cosmology to derive from it.  I went the other direction.  But, it was Cosmology after all.  His particle values were for (today) -- not 10 Billion years ago.  I can scale all his values rather easily.  But I dread the thought of what must happen to epsilon and mu, though still epsilon*mu = 1/c^2.  Some poor scientist must reintroduce these into Schrodinger's Equations.  Then see how they effect electron orbits and space gas dynamics.  Type I Supernovas noticed "Dark Energy" which might merely come from changes in epsilon effecting changes in electron cloud sizes -- and nothing more.
A: To change your system of units so that some constants, like $\epsilon_{0}$ and $\mu_{0}$, disappear (are set to 1) changes the VALUE of the constants but not their RELEVANCE.  These two constants determine the constants of proportionality of the electric field and the magnetic field and their relationship to the speed of light, in ANY system of units, will always exist — be physical.
