Oscillations in forces other than the Weak As I understand it neutrino oscillations arise due to the neutrino mass eigenstates being distinct from the neutrino flavour eigenstates.
Flavour eigenstates are the states in which neutrinos interact via the weak force, and so are the eigenstates in which they are created and detected, and consist of a superposition of the the mass eigenstates. The mass eigenstates correspond to how the neutrino propagates through spacetime. The finite neutrino mass causes the relative phase of each mass eigenstate to change as the neutrino propagates, changing the probability that it will be detected in a given flavour eigenstate (hence an electron neutrino can oscillate into a muon neutrino and so on).
My question is, can other forces exhibit similar oscillations? For example, in principle is it possible for an electron to oscillate it's charge state as it travels, to be detected as a positron some distance later? Or are charge eigenstates not the similarly related to mass eigenstates?
 A: Think that there is some confusion. I assume that you associate particle oscillations with the weak interaction because you consider the specific example of the neutrinos (that indeed can only weakly interact)
See also the wikipedia article about neutral particle mixing:
You need a particle $A$, that is different from its antiparticle $B$, but with the same decay-products $F$. You then get the possibility for oscillations:
$$A  →  F  →  B  →  F  →  A  →  ...$$

More concretely: it is also experimentally established that neutral kaons oscillates.
Kaons are produced by the strong interaction in eigenstates
$$\overline{K}^0, K^0$$ 
but they decay by the weak interaction as CP-eigenstates
$$K_1,K_2.$$ 
Since quark flavour is not conserved in weak interactions transitions between neutral mesons and their particles are possible.
What is observed is that:

An initially pure beam of $K_0$ will turn into its antiparticle while propagating, which will turn back into the original particle, and so on. (Wikipedia)

A: Only weak interactions can change the flavor of quarks and, therefore, produce mixing (oscillation) effects in mesons. Mixing always involves the weak interaction because it requires to change the flavor of the constituent quarks (strong and electromagnetic interactions can not change flavor).
Also, electric charge is conserved in all interactions and, therefore, mixing can only affect neutral particles.
