The space-air-ground-sea integrated network (SAGSIN) plays an important role in offering global coverage. To improve the efficient utilization of spectral and hardware resources in the SAGSIN, integrated sensing and communications (ISAC) has drawn extensive attention. Most existing ISAC works focus on terrestrial networks and cannot be straightforwardly applied in satellite systems due to the significantly different electromagnetic wave propagation properties. In this work, we investigate the application of ISAC in massive multiple-input multiple-output (MIMO) low earth orbit (LEO) satellite systems. We first characterize the statistical wave propagation properties by considering beam squint effects. Based on this analysis, we propose a beam squint-aware ISAC technique for hybrid analog/digital massive MIMO LEO satellite systems exploiting statistical channel state information. Simulation results demonstrate that the proposed scheme can operate both the wireless communications and the target sensing simultaneously with satisfactory performance, and the beam-squint effects can be efficiently mitigated with the proposed method in typical LEO satellite systems.