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Performing neural network inference on encrypted data without decryption is one popular method to enable privacy-preserving neural networks (PNet) as a service. Compared with regular neural networks deployed for machine-learning-as-a-service, PNet requires additional encoding, e.g., quantized-precision numbers, and polynomial activation. Encrypted input also introduces novel challenges such as adversarial robustness and security. To the best of our knowledge, we are the first to study questions including (i) Whether PNet is more robust against adversarial inputs than regular neural networks? (ii) How to design a robust PNet given the encrypted input without decryption? We propose PNet-Attack to generate black-box adversarial examples that can successfully attack PNet in both target and untarget manners. The attack results show that PNet robustness against adversarial inputs needs to be improved. This is not a trivial task because the PNet model owner does not have access to the plaintext of the input values, which prevents the application of existing detection and defense methods such as input tuning, model normalization, and adversarial training. To tackle this challenge, we propose a new fast and accurate noise insertion method, called RPNet, to design Robust and Private Neural Networks. Our comprehensive experiments show that PNet-Attack reduces at least $2.5\times$ queries than prior works. We theoretically analyze our RPNet methods and demonstrate that RPNet can decrease $\sim 91.88\%$ attack success rate.