Deformation and breakup of viscoelastic drops translating through a Newtonian fluid within capillaries of different cross-sections is experimentally examined. Capillaries with circular, square, and non-uniform cross-sections are considered, and the effect of the confining geometry on breakup behavior is examined. In particular, the various modes of drop breakup, as well as the critical conditions for the onset of each mode of breakup, are identified for each type of confining geometry. In the case of pressure-driven motion, large drops become unstable as the strength of the imposed flow increases. Three different modes of drop breakup are identified. At low polymer concentrations, large drops develop a region of negative curvature at their trailing ends, resulting in a mode of breakup that is qualitatively similar to the re-entrant jet mechanism observed experimentally (and predicted numerically) for large Newtonian drops. As the polymer concentration is increased, the re-entrant cavity at the trailing end of large drops develops a cusp within the drop which eventually leads to breakup through tip-streaming with increasing capillary number. At even higher polymer concentrations, increasing the capillary number leads to breakup through rim-streaming, where a stream of small droplets is continuously ejected from several points along the trailing rim (on or near the rear stagnation ring). The rim-streaming mode of breakup is only observed in the cylindrical capillary. For both cylindrical and square capillaries, the critical capillary number for the onset of drop breakup is a decreasing function of drop size, with the critical capillary numbers for the onset of cusp streaming in the square capillary being slightly smaller than those in the cylindrical capillary.