The right hand rule has no actual physical meaning. It's simply a sign convention (i.e. that (\vec{A} \times \vec{B})_x = A_yB_z - A_zB_y rather than A_zB_y - A_yB_z, and so on). This amounts to an ambiguity in direction for certain quantities. At a formal mathematical level, what we can say is that of the three quantities involved in a cross-product, only two can be normal vectors. The third must be what's known as an "axial vector" or "pseudovector."
In the case of the Lorentz force law, the pseudovector is the magnetic field, since we already know there's ambiguity in the direction of a velocity or a force. Since the magnetic field is a pseudovector, it should be possible to construct it as a cross-product between two normal vectors. (Or, objects that transform like normal - or "polar" - vectors.) And, in fact, we have such a notion. The magnetic vector potential is a normal vector. So, when we write \vec{B} = \vec{\nabla} \times \vec{A}, we introduce a quantity that lacks the sign ambiguity. (You may want to try writing the Lorentz force law in terms of \phi and \vec{A}, to convince yourself that the right hand rule is actually irrelevant here.)
From the point of view of quantum physics, the vector and scalar potentials are the fundamental EM degrees of freedom; and, the only ambiguity they have is gauge invariance, which is (in a certain sense) seem as a feature of the theory.
