Electrons obey the right hand rule when a magnetic field bends their path. According to the right-hand rule, will positrons bend in the same direction?
The magnetic force is $$ \vec F = q \vec v \times \vec B $$ so with a constant $\vec B, \vec v$, it's clear that the opposite sign of $q$ leads to the opposite force. So antiparticles surely bend in the opposite direction than the original particles.
Concerning your "right hand rule", I think that whichever you meant, it was misintetrpreted. Right hand rules aren't satisfied by individual particle species. They're general for all charged particles etc. At any rate, if you have a version of the right-hand rule that applies to electron's acceleration, the rule for positrons is the other one.
A nice bubble chamber picture:
"Electrons and positrons produced simultaneously from individual gamma rays curl in opposite directions in the magnetic field of a bubble chamber. In the above example the gamma ray has lost some energy to an atomic electron, which leaves the long track, curling left. The gamma rays do not leave tracks in the chamber, as they have no electric charge"