Fluid flows in microfluidic devices are often characterized by non-Newtonian rheology with non-linear wall slip behavior also observed. This work solves this problem class with the lattice Boltzmann method (LBM), proposing new advanced boundary scheme formulations to model the joint contribution of non-linear rheology and non-linear wall slip laws in application to microchannels of planar and circular cross-section. The non-linear stress-strain-rate relationship of the microflow is described by a generalized Newtonian model where the viscosity function follows the Sisko model. To guarantee that LBM steady-state solutions are not contaminated by numerical errors that depend on the viscosity local value, the two-relaxation-time (TRT) collision is adopted. The fluid-wall accommodation model considers different slip laws, such as the Navier linear, Navier non-linear, empirical asymptotic and Hatzikiriakos slip laws. They are transcribed into the LBM framework by adapting the local second-order boundary (LSOB) scheme strategy to this problem class. Theoretical and numerical analyses developed for a steady and slow viscous fluid within 2D slit and 3D circular pipe channels demonstrate the parabolic level of accuracy of the developed LSOB scheme throughout the considered non-linear slip and non-Newtonian models.