Electron-acoustic shock waves (EASWs) in an unmagnetized electron-positron-ion plasma system (consisting of a cold mobile viscous electron fluid, hot electrons and positrons following the q-nonextensive distribution, and immobile positive ions) are studied analytically. The Burgers equation is derived by using the well-known reductive perturbation method. The basic features (viz. polarity, amplitude, width, phase speed, etc.) of EASWs are briefly addressed. The basic features of EASWs are found to be significantly modified by the effects of nonextensivity of the hot electrons and positrons, the relative number density and temperature ratios, and the kinematic viscosity of the cold electrons. The present investigation can be useful in understanding the fundamental characteristics of EASWs in various space plasmas.