In the field of medical image compression, Implicit Neural Representation (INR) networks have shown remarkable versatility due to their flexible compression ratios, yet they are constrained by a one-to-one fitting approach that results in lengthy encoding times. Our novel method, ``\textbf{UniCompress}'', innovatively extends the compression capabilities of INR by being the first to compress multiple medical data blocks using a single INR network. By employing wavelet transforms and quantization, we introduce a codebook containing frequency domain information as a prior input to the INR network. This enhances the representational power of INR and provides distinctive conditioning for different image blocks. Furthermore, our research introduces a new technique for the knowledge distillation of implicit representations, simplifying complex model knowledge into more manageable formats to improve compression ratios. Extensive testing on CT and electron microscopy (EM) datasets has demonstrated that UniCompress outperforms traditional INR methods and commercial compression solutions like HEVC, especially in complex and high compression scenarios. Notably, compared to existing INR techniques, UniCompress achieves a 4$\sim$5 times increase in compression speed, marking a significant advancement in the field of medical image compression. Codes will be publicly available.