How to prove that both $\mu_0$ and $\epsilon_0$ don't depend on any frame of references?

The way to this is to make the hypothesis at the outset that $\epsilon_0$ and $\mu_0$ are scalar invariant constants, and then check whether, under this hypothesis, the equations overall are Lorentz covariant. It turns out that they are. But it is easier to prove this by starting out with tensor notation which I am guessing you have not learned yet.

Now I will unpack the terminology used above.

  1. scalar = fully specified at each event by a single number

  2. invariant = the number you get is the same in all reference frames

  3. constant = the number you get is the same at all events in any given reference frame

So one is claiming quite a lot for these $\epsilon_0$ and $\mu_0$. It amounts to claiming that they are just numbers like 2 and $\pi$, except that they may have physical dimensions in the system of units being adopted. Having made the claim, the logic is, as I already said, that one now asks whether, if these quantities are indeed scalar invariant constants, then do the Maxwell equations survive unchanged from one frame to another? One can prove that they do by a rather lengthy calculation involving the transformation of force, or by a quicker calculation involving tensors.

(I expound this point fully and carefully in my own book on this subject; it is an undergraduate physics textbook.)

Maxwell DID NOT assume that $\mu_0$ and $\epsilon_0$ are constant. They were invented by an Italian engineer named Georgi much later. They have little to do with physics, so they won't change. $\mu_0/4\pi=10^{-7}$ is a conversion constant from physical units to SI. $10^{-5}$ comes from converting cgs to MKS. $10^{-2}$ comes from redefining the Ampere in the 1880's. $1/4\pi\epsilon_0$ is just $c^2$ in converted units.