So if the mass eigenstates aren't different then you don't get mixing between the flavor eigenstates, so this is expected? And by measuring the frequency of the FCNCs then that is what gives them the difference between the mass eigenstates? (bear with me, its been 25 years since I decided to become a programmer instead...)

Δm is directly proportional to the frequency of the oscillation. (In natural units it is exactly the angular frequency: cos(Δm·t) appears in the decay rate) So indeed Δm=0 means no oscillation. You measure it by essentially counting the number of particle and antiparticle decays as a function of decay-time.

What about charged systems like proton-antiproton? We expect there to be some similarly tiny Δm there, but obviously you can't have FCNC because its no longer neutral?

Protons and antiprotons must have the same mass under CPT symmetry