  We study nanodevices based on ultrathin superconducting nanowires connected
in parallel to form nanowire SQUIDs. The function of the critical current
versus magnetic field, $I_{C}(B)$, is multivalued, asymmetric and its maxima
and minima are shifted from the usual integer and half integer flux quantum
points. The nanowire interference device is qualitatively distinct from
conventional SQUIDs because nanowires do not obey the same current-phase
relationship (CPR) as Josephson junctions. We demonstrate that the results can
be explained assuming that (i) the CPR is linear and (ii) that each wire is
characterized by a sample-specific critical phase, which is usually much larger
than $\pi/2$. Our proposed model offers accurate fits to $I_{C}(B)$. It
explains the single-valuedness regions where only one vorticity (i.e., the
order parameter winding number) is stable as well as regions where multiple
vorticity values are allowed for the SQUIDs. We also observe and explain
regions in which the standard deviation of the switching current is independent
of the magnetic field. We develop a technique that allows a reliable detection
of hidden phase-slips. Using this technique we find that our model correctly
predicts the boundaries of vorticity regions, even at low currents where
$I_C(B)$ is not directly measurable.
