In addition to the quantities [tex]\mu_0[/tex] and [tex]\epsilon_0[/tex], the only dimensionful units in the equations Matthew posted (and in Maxwell's equations + Lorentz force, which describe more fully how the electric and magnetic forces interact with charges) have units of: **charge, distance** and **time**.

You certainly can redefine your unit of charge to eliminate either [tex]\mu_0[/tex] or[tex]\epsilon_0[/tex]. To eliminate both, you need to redefine your measure of space or time, which is simply choosing a longer/shorter measuring stick or choosing to measure units of time that are longer/shorter than 1 second. Doing so doesn't change the speed of light, it just changes the units you're using to measure it.