We propose a novel predictive lossless compression algorithm for regions of interest (ROIs) in the hyperspectral images via Maximum Correntropy Criterion (MCC) based Least Mean Square (LMS) filtering. Non-linearity and non-Gaussian conditions of prediction residuals of the ROI pixels in the hyper-spectral image are taken into account to improve the compression performance compared to the ordinary LMS used in the Consultative Committee for Space Data Systems (CCSDS) standard. Test results on hyperspectral image datasets show that the proposed method outperforms several other state-of-the-art methods.