The deformation of microfluidic channels in a soft elastic medium has a central role in the operation of lab-on-a-chip devices, fluidic soft robots, liquid metal (LM) electronics, and other emerging soft-matter technologies. Understanding the influence of mechanical load on changes in channel cross section is essential for designing systems that either avoid channel collapse or exploit such collapse to control fluid flow and connectivity. In this paper, we examine the deformation of microchannel cross sections under far-field compressive stress and derive a "gauge factor" that relates externally applied pressure with change in cross-sectional area. We treat the surrounding elastomer as a Hookean solid and use two-dimensional plane strain elasticity, which has previously been shown to predict microchannel deformations that are in good agreement with experimental measurements. Numerical solutions to the governing Lame (Navier) equations are found to match both the analytic solutions obtained from a complex stress function and closed-form algebraic approximations based on linear superposition. The application of this theory to soft microfluidics is demonstrated for several representative channel geometries.