How a moving electron gets deflected in a magnetic field in on direction and the positron gets deflected in the opposite direction? In a question Why does nature prefer the right-hand rule over the left-hand rule? was another question

My question is, why this direction and not the opposite?

This part of the post received no attention and so I ask it here explicitly.
 A: You are correct that this part of the question was not addressed, but you did not quote the context:

If you have a coil in which you put some current, and the nature creates a magnetic field in a direction perpendicular to it, the direction depends on the right hand rule.
My question is, why this direction and not the opposite?

Here we hit on one of the basic problems of the standard model of physics, that our universe is made out of baryons. There is no symmetry in the existing matter in the universe between baryons and antibaryons, but our models have no adequate explanation why this happens. It leads to CP violation, but the standard model does not have enough of it to model  why matter, atoms, molecules solids liquids and gas in the universe are mainly matter and not antimatter  .
One can understand that a solid earth cannot be made of equal numbers of matter and antimatter, because almost everything would annihilate and turn finally into photons and neutrinos.  So without addressing the cosmological problem, let us look at earth as it is. It is made out of baryons, protons and neutrons which in nuclei gather electrons to become neutral.
This means that the nuclei are positive and the leptons around them negative. One could define them the other way, but that is a linguistic definition. So even though Maxwell's equations have a symmetry between positive and negative and the definition of right and left is just that, a definition, when generating currents in matter, there is no choice other than naming plus or minus. Since the protons are positive and the electrons negative, the right hand rule is defined by the way the current is generated, and in conductors the current is generated by the drifting electrons. There is no symmetry here, except in special experiments where positive ions generate the current.
For your general title:

How a moving electron gets deflected in a magnetic field in one direction and the positron gets deflected in the opposite direction?

the answers in the question you link should be adequate, Maxwell's equations describe a symmetric behavior, the charge is irrelevant if one only has electrons and positrons. It is in interactions with atoms and nuclei that a difference exists.
