The equilibrium states of a thin-walled hollow cylinder containing a 'freak link", (e .g. a Josephson junction ) are examined in a Bose gas model calculation. It is found that these states may be such that the fluxoid is not an integer multiple of the flux quantum. The deviation from integer flux quantization depends upon the coupling strength of the "weak link" and the externally applied magnetic flux. The first order reduced density matrix (the 1-matrix) for the boson system is then investigated utilizing the general features of the model calculation. The fluxoid quantization and persistent currents are considered in terms of the modifications (due to the link) of the off-diagonal long- ra nge order of the 1-ma tr ix .