In this manuscript, we address the long-standing question of whether a single theory for model plastic fluids is suitable to deal with the unidirectional compression problem in magnetorheological (MR) fluids. We present an extensive experimental investigation of the performance of MR fluids in slow-compression, no-slip, constant-volume squeeze mode under different magnetic field strengths (0-354 kA/m), dispersing medium viscosities (20-500 mPa center dot s) and particle concentrations (5-30 vol%). Normal force versus compressive strain curves reasonably collapse when normalizing by the low-strain normal force. Deviations from the squeeze flow theory for field-responsive yield stress fluids are associated to microstructural rearrangements under compression in good agreement with the so-called squeeze strengthening effect. Yield compressive stresses are found to scale as H-2.0.