The present work addresses to analyse the heat transfer enhancement of unsteady laminar incompressible MHD flow of couple stress nanofluid through a rectangular squeezing channel with thermal radiation. The governing transport equations are reduced into an ordinary differential equation using relevant similar transformations and then solved numerically with the shooting method Runge–Kutta fourth-order scheme. The results are sketched as two-dimensional graphs and explained in detail for flow and heat transfer with non-dimensional physical parameters. Further calculated the skin friction and heat transfer rates at the upper plate. The axial velocity of engine oil/ethylene glycol has been increased by β and Da−1 towards the centre of the channel decreased for squeezing, and it is opposite for the separation case. The numerical results have good agreement with published work for temperature profile. © 2020, Akadémiai Kiadó, Budapest, Hungary.