The liquid metal eutectic gallium-indium (EGaIn) is useful for forming soft electrodes (for characterizing charge transport through self-assembled monolayers) and for forming stable electronic structures (e.g., electrodes, heaters, electromagnets) in microchannels. EGaIn is well-suited for these applications because of its rheological properties at room temperature: it behaves like an elastic material until it experiences a critical surface stress, at which point it yields and flows readily. These properties allow EGaIn to fill microchannels rapidly when sufficient pressure is applied to the inlet of the channels, yet maintain structural stability within the channels once ambient pressure is restored. Experiments conducted in microfluidic channels, and in a parallel-plate rheometer, suggest that EGaIn's behavior is dictated by the properties of its surface (predominantly gallium oxide, as determined by Auger measurements); these two experiments both yield approximately the same number for the critical surface stress required to induce EGaIn to flow (~0.5 N/m). This analysis–which shows that the pressure that must be exceeded for EGaIn to flow through a microchannel is inversely proportional to the critical (i.e., smallest) dimension of the channel–is useful to guide future fabrication of microfluidic channels to mold EGaIn into functional microstructures.