The force to squeeze a Herschel - Bulkley material without slip between two approaching surfaces of various curvature is calculated. The Herschel - Bulkley yield stress requires an infinite force to make plane - plane and plane - concave surfaces touch. However, for plane -convex surfaces this force is finite, which suggests experiments to access the mesoscopic thickness region ( 1 - 100 mu m) of non-Newtonian materials using squeeze flow between a plate and a convex lens. Compared to the plane - parallel surfaces that are used most often for squeeze flow, the dependence of the separation h' and approach speed V on the squeezing-time is more complicated. However, when the surfaces become close, a simplification occurs and the near-contact approach speed is found to vary as V v proportional to h'(0) if the Herschel -Bulkley index is n< 1/3, and V v proportional to h'((3n-1)/(2n)) if n >= 1/3. Using both plane - plane and plane - convex surfaces, concordant measurements are made of the Herschel - Bulkley index n and yield stress tau(0) for two soft solids. Good agreement is also found between tau(0) measured by the vane and by each squeeze-flow method. However, one of the materials shows a limiting separation and a V( h') behaviour not predicted by theory for h' < 10 mu m, possibly owing to an interparticle structure of similar lengthscale.