The effects of polymer molecular weight (MW) and elasticity on filament thinning and breakup are investigated in microchannel cross flow. When a viscous solution is stretched by an external immiscible fluid, dilute high MW polymeric solutions strongly affect the breakup process, compared to the Newtonian case of the same shear viscosity. At late times when viscoelastic stresses become important, polymer filaments show much slower evolution, morphology featuring multiple connected drops, and different scaling with the ratio of flow rates. These effects diminish monotonically but nonlinearly as the polymer MW is decreased. For all fluids, we find two regimes that govern filament thinning: a flow-driven and a capillary-driven regime. In the flow-driven regime, the filament thinning process can be described in terms of extensional viscosities of the immiscible fluids, which for the polymeric solutions includes strain hardening.