Superconducting quantum interference devices (SQUIDs) that incorporate two superconductor/insulator/superconductor (SIS) Josephson junctions in a closed loop form the core of some of the most sensitive detectors of magnetic and electric fields currently available. [1,2] SQUIDs in these applications are typically operated with a finite voltage which generates microwave radiation through the ac Josephson effect. This radiation may impact the system being measured. We describe here a SQUID in which the Josephson junctions are formed from strips of normal metal (N) in good electrical contact with the superconductor (S). Such SNS SQUIDs can be operated under a finite voltage bias with performance comparable or potentially better than conventional SIS SQUIDs. However, they also permit a novel mode of operation that is based on the unusual interplay of quasiparticle currents and supercurrents in the normal metal of the Josephson junction. The new method allows measurements of the flux dependence of the critical current of the SNS SQUID without applying a finite voltage bias across the SNS junction, enabling sensitive flux detection without generating microwave radiation.