  Full-shell nanowires (a semiconducting core fully wrapped by an epitaxial
superconducting shell) have recently been introduced as promising hybrid
quantum devices. Despite this, however, their properties when forming a
Josephson junction (JJ) have not been elucidated yet. We here fill this void by
theoretically studying the physics of JJs based on full-shell nanowires. In the
hollow-core limit, where the thickness of the semiconducting layer can be
ignored, we demonstrate that the critical supercurrent $I^{c}$ can be tuned by
an external magnetic flux $\Phi$. Specifically, $I^{c}(\Phi)$ does not follow
the Little-Parks modulation of the superconducting pairing ${\Delta}(\Phi)$,
and exhibits steps for realistic values of nanowire radii. The position of the
steps can be understood from the underlying symmetries of the orbital
transverse channels which contribute to the supercurrent for a given chemical
potential.
