Recently, large language models (LLMs) have driven promising progress in lossless image compression. However, directly adopting existing paradigms for medical images suffers from an unsatisfactory trade-off between compression performance and efficiency. Moreover, existing LLM-based compressors often overlook the security of the compression process, which is critical in modern medical scenarios. To this end, we propose a novel joint lossless compression and steganography framework. Inspired by bit plane slicing (BPS), we find it feasible to securely embed privacy messages into medical images in an invisible manner. Based on this insight, an adaptive modalities decomposition strategy is first devised to partition the entire image into two segments, providing global and local modalities for subsequent dual-path lossless compression. During this dual-path stage, we innovatively propose a segmented message steganography algorithm within the local modality path to ensure the security of the compression process. Coupled with the proposed anatomical priors-based low-rank adaptation (A-LoRA) fine-tuning strategy, extensive experimental results demonstrate the superiority of our proposed method in terms of compression ratios, efficiency, and security. The source code will be made publicly available.