We study magneto-transport through topological insulator nanowires shaped in the form of a
constriction, as can be obtained by etching techniques. The magnetic field is coaxial, potentially
turning the nanowire into a magneto-chiral junction. We show in a detailed analytical and numerical
study that two main transport regimes emerge, depending on the central narrow region being short
or long as compared to the magnetic length at the junction entrance and exit. In both cases the
central region hosts Dirac-particle-in-a-box states due to magnetic confinement, whose conductance
properties are strongly influenced by Landau levels at the ends of the constriction. Notably, in
the low-energy regime only chiral states with a specific handedness can transport charge across the
junction. Based on these properties and general symmetry considerations we argue that the shaped
nanowire should exhibit strong magneto-chiral non-reciprocal transport beyond linear response. We
employ a numerical tight-binding implementation of an effective 2D model on a non-homogeneous
grid, capable of simulating samples of realistic sizes, and test its soundness against full simulations
for scaled-down 3D topological insulator wires.