  We use a model of hard-core bosons to describe a SQUID built with two
crystals of $d_{x^2-y^2}$-superconductors with orientations (100) and (110).
Across the two faceted (100)/(110) interfaces, the structure of the
superconducting order parameter leads to an alternating sign of the local
Josephson coupling, and the possibility of quartet formation. Using a mapping
of the boson model to an $XXZ$ model, we calculate numerically the energy of
the system as a function of the applied magnetic flux, finding signals of
$hc/4e$ oscillations in a certain region of parameters. This region has a large
overlap to that at which binding of bosons exists.
