As your link says, in non-magnetic materials (most of them!), the magnetic field of each atom is disorganized, so it goes in every direction and sums up to a near zero result.

Magnets are more organized, so they do have a magnetic field as a whole.

So what would happen to the non magnetic atoms if they had too much of external magnetic force directed at them? Whilst they dont repel or attract like magnetic atoms, they must be influenced repelled or attracted in some way by a really strong external magnetic force (like this article is suggesting) or would/could it break apart the orbiting electrons?

An opposing magnetic field is induced, which can cause the object to float if strong enough: https://en.m.wikipedia.org/wiki/Diamagnetism

I think he is referring to microscopic magnetic imperfections in the atoms. It's my understanding these can be exploited to apply force on the atom, but we don't know how. Useful theoretically as a reaction steering mechanism.

What does it mean to be organized? Is magnetism just a vector with an unknown source?

The cumulative effect of majority direction alignment

Sure, but where does that direction come from and why does it matter that there's majority?