Slow pulsed capillary discharges in round capillaries are currently under investigation for use as plasma channel laser waveguides in laser-wakefield acceleration, x-ray lasers, and higher-harmonic generation. In this study, a capillary discharge with a square cross section is presented. The electron density, which determines the laser guiding properties, can be measured by means of transverse interferometry in this device. Using a numerical model of the plasma and the capillary wall, an analysis of the discharge is made. The results predict that the square channel is capable of guiding circular laser pulses. The guiding properties are quite similar to those of a round channel with nearly the same diameter as the channel width. This suggests the results obtained by measuring the square capillary discharge are applicable for round channels as well. It was found that the wall heating was inhomogeneous, which makes the wall more susceptible to ablation. The heating of the wall changes the transverse optical pathlength in the interferometry experiments.