Why does magnetism penetrate all known materials

ACuriousMind is of course correct, in the area of superconducting materials, but for the rest of the material world, I have always thought it was because magnetic monopoles, as far we know, cannot exist.

You have the mathematical basis for this in your post, re the Maxwell equation quoted.

All magnetic field lines therefore have to begin and end on some particular material surface.

Since electrons have magnetic dipole moments an external field align this moments and form a magnetic field inside every material.

BTW the saturation of any material happens due to the finite number of available electrons which could be aligned.

The mentioned by ACuriousMind Messner effect takes place because according to Cooper the electrons unite to couples. By this they will be aligned with their opposite magnetic dipole moments and form a magnetic loop and for the external field it is easier to flow around the material instead of going through. This happens only at temperatures near 0 Kelvin, where the movement of electrons is strongly inhibited an Cooper pairs will be stable. And behind a certain strength of the external magnetic field the Cooper pairs will be destroyed.

I think one should first reframe the question and ask, how do you prevent the penetration of magnetic fields in a solid in the first place? As noted in another answer, there are no magnetic monopoles in nature, so the only possible way* is to block the field is to run a current in the material which generates a field that counteracts the applied magnetic field.

If the only way for a material to to block an external magnetic field is through a current, we should think about how that current can be sustained in time. With the exception of superconductors, all materials have nonzero resistance. Therefore, the current needed to block penetration of magnetic fields will always decay in time. Thus, in general, there is no way to screen static magnetic fields in a material with nonzero resistance.

On the other hand, for alternating magnetic fields, most metals block magnetic fields just fine (provided they are thick compared to the skin depth) and the magnetic field does not penetrate except in an exponentially suppressed fashion. The key point here is Faraday's law which tells you that you need a changing magnetic field to generate a current.

*Note: You could also utilize the spin magnetic moment of elementary particles, but this fails for large magnetic fields that OP cares about (above 1 Tesla) except perhaps in neutron stars where the density is very high. Methods using mu metal and ordinary diamagnetism fall under this category.