The steady-state fully developed laminar flow of non-Newtonian power-law fluids is analytically studied in a circular microchannel under an imposed uniform and constant wall heat flux. Increasing the flow behavior index results in broadening the dimensionless temperature distribution, i.e., in enlarging the wall and bulk fluid temperature difference. Similar behavior may also be observed when heating or cooling flux is reduced. For any particular value of the flow behavior index, a critical Brinkman number exists in which the bulk mean fluid temperature equals the wall temperature; in this special case of surface cooling, the Nusselt number tends to infinity. Dilatants (shear-thickening fluids) demonstrate more tangible reactions than pseudoplastics (shear-thinning fluids) to changes in the Brinkman number. Entropy generation increases with the flow behavior index as well as the Brinkman number. For shear-thickening fluids, the entropy generation rate from heat transfer is more than the entropy generation rate from fluid friction, while an opposite trend is observed for shear-thinning fluids.