Medical image registration is a crucial process for aligning anatomical structures, enabling applications such as atlas mapping, longitudinal analysis, and multimodal data fusion. This paper introduces DINO-Reg, an adaptation-free registration method leveraging the vision foundation model, DINOv2, to extract features for deformable 3D medical image alignment. Although DINOv2 was originally trained on natural images, our study links the vision foundation model with medical image registration and demonstrates that the generic image encoder could readily generalize to medical images with state-of-the-art performance. We further propose DINO-Reg-Eco, a knowledge-distilled version using a UNet-structured 3D convolutional neural network (CNN) for feature extraction. The Eco model reduces encoding time by 99% while maintaining state-of-the-art performance, which is essential for resource-limited settings and significantly lowers the carbon footprint associated with intensive computational demands. Benchmarking across diverse datasets shows that both methods outperform existing supervised and unsupervised approaches without fine-tuning, demonstrating the transformative potential of foundation models in medical image registration. Our code is open-sourced at https://github.com/RPIDIAL/DINO-Reg.