The substantial progress in privacy-preserving machine learning (PPML) facilitates outsourced medical computer-aided diagnosis (MedCADx) services. However, existing PPML frameworks primarily concentrate on enhancing the efficiency of prediction services, without exploration into diverse medical services such as medical segmentation. In this paper, we propose MedShield, a pioneering cryptographic framework for diverse MedCADx services (i.e., multi-service, including medical imaging prediction and segmentation). Based on a client-server (two-party) setting, MedShield efficiently protects medical records and neural network models without fully outsourcing. To execute multi-service securely and efficiently, our technical contributions include: 1) optimizing computational complexity of matrix multiplications for linear layers at the expense of free additions/subtractions; 2) introducing a secure most significant bit protocol with crypto-friendly activations to enhance the efficiency of non-linear layers; 3) presenting a novel layer for upscaling low-resolution feature maps to support multi-service scenarios in practical MedCADx. We conduct a rigorous security analysis and extensive evaluations on benchmarks (MNIST and CIFAR-10) and real medical records (breast cancer, liver disease, COVID-19, and bladder cancer) for various services. Experimental results demonstrate that MedShield achieves up to 2.4×, 4.3×, and 2× speed up for MNIST, CIFAR-10, and medical datasets, respectively, compared with prior work when conducting prediction services. For segmentation services, MedShield preserves the precision of the unprotected version, showing a 1.23% accuracy improvement.