Cryptographically secure neural network inference typically relies on secure computing techniques such as Secure Multi-Party Computation (MPC), enabling cloud servers to process client inputs without decrypting them. Although prior privacy-preserving inference systems co-design network optimizations with MPC, they remain slow and costly, limiting real-world deployment. A major bottleneck is their use of a single, fixed transformer model for all encrypted inputs, ignoring that different inputs require different model sizes to balance efficiency and accuracy. We present SecureRouter, an end-to-end encrypted routing and inference framework that accelerates secure transformer inference through input-adaptive model selection under encryption. SecureRouter establishes a unified encrypted pipeline that integrates a secure router with an MPC-optimized model pool, enabling coordinated routing, inference, and protocol execution while preserving full data and model confidentiality. The framework includes training-phase and inference-phase components: an MPC-cost-aware secure router that predicts per-model utility and cost from encrypted features, and an MPC-optimized model pool whose architectures and quantization schemes are co-trained to minimize MPC communication and computation overhead. Compared to prior work, SecureRouter achieves a latency reduction by 1.95x with negligible accuracy loss, offering a practical path toward scalable and efficient secure AI inference. Our open-source implementation is available at: https://github.com/UCF-ML-Research/SecureRouter