In the contemporary digital healthcare setting, the dissemination and sharing of medical imagery are integral to routine medical operations. However, medical images often contain sensitive patient information, and without adequate protection, there is a risk of illegal acquisition or leakage, which brings unnecessary troubles. To address this issue, an encryption algorithm based on Zigzag scrambling and a new four-dimensional hyperchaotic system was proposed. Firstly, the Zigzag algorithm was used to scramble the image once, roughly hiding the obvious contours of the image. Then, an improved cat mapping algorithm was used to perform secondary scrambling on the image, removing obvious texture features. Finally, the scrambling factor generated from the plaintext image was applied to the initial value generation process of the hyperchaotic system. The generated hyperchaotic sequence was transformed into a hyperchaotic matrix for the subsequent diffusion process of the encryption algorithm. The simulation results show that the proposed algorithm can effectively conceal plaintext information based on the characteristics of medical images and resist common types of attacks. The robustness of the proposed algorithm has been demonstrated through testing, confirming its capability to address the issue of image interference in remote healthcare.