Should a black hole, in the absence of all external electromagnetic forces, generate its own magnetic field? Would all particles in the vicinity of the black hole by effected by this magnetic field--i.e., are there any particles that do not respond to some magnetic force?
A non-rotating black hole can have an electric field, but the difference between electric and magnetic fields depends on the restframe. An electric field in motion generates a magnetic field and vice-versa. Hence, black holes can be said to have magnetic fields, in particular if they rotate. Electric and magnetic fields always come together.
You may instead ask why a black hole has electric charge but no magnetic charge. That's because in Maxwell's theory of electrodynamics, there are no magnetic monopoles. If there were magnetic monolopes, black holes could also have magnetic charge. (These things are known as 'dyonic black holes'.)
As to why the electromagnetic field of a black hole has the form it has, that's simply what you get if you solve the Maxwell-equations coupled to General Relativity.
The definition of a black hole describes a body that's information is hidden behind an event horizon if the black hole has a magnetic field the field must be limited to within the event horizon as otherwise the field could have a causal effect on events outside the horizon and violate the definition