Is there a way to understand why a substance would be diamagnetic, paramagnetic, or ferromagnetic without knowledge of quantum mechanics? That is, can these phenomena be explained without quantum mechanics?
Diamagnetism occurs in all elements for a similar reason to Lenz's law, the magnetic field applied to the substance, induces electric currents which produce a magnetic field that opposes the applied field. The reason that diamagnetism is only observed in atoms where electrons are paired, is because then any paramagnetic effects are zero and you can observe the underlying diamagnetism, which would be swamped by any paramagnetic effects if there were unpaired electrons.
The existence and stability of atoms cannot even be explained classically, so in that sense the answer is no.
If you allow a semiclassical model where you forget about radiative losses and imagine the electron in orbit around the nucleus, then you can kind of account for diamagnetism. If you add that the spin angular momentum of the electron can only take two values as measured along any axis, then you can more or less account for paramagnetism.
Ferromagnetism, however, requires the quantum mechanical concept of exchange interaction to explain, and this has no classical analogue.
All in all, magnetism is a pretty deeply quantum mechanical phenomenon. Explaining it "classically" (to the extent that's even possible) requires us to add in a bunch of quantum mechanical results as ad hoc rules, so the real answer to your question is no.
At a basic level , diamagnetism occurs in those compounds / element which have all electrons paired.
Eg in $H_2 $all electrons in Bonding Orbital are paired
On the other hand paramagnetism occurs in those compounds / element which don't have all electrons paired.
Eg in $O_2$ , 2 electrons are unpaired in the anti-bonding molecular orbital
It can similarly be applied on others too.