A classical result of Colliot-Thélène and Sansuc states that the only obstruction to the Hasse principle and weak approximation for generalised Châtelet surfaces is the Brauer-Manin one, conditionally on Schinzel's hypothesis. Inspired by their work, we study the analogous questions concerning the existence and the density of integral points on the corresponding affine surfaces, again under Schinzel's hypothesis. To be precise, we show that the Brauer-Manin obstruction is the only obstruction to the integral Hasse principle for an infinite family of generalised affine Châtelet surfaces. Moreover, we show that the set of integral points on any surface in this family satisfies a strong approximation property off infinity with Brauer-Manin obstruction.