Balancing privacy and predictive utility remains a central challenge for machine learning in healthcare. In this paper, we develop Syfer, a neural obfuscation method to protect against re-identification attacks. Syfer composes trained layers with random neural networks to encode the original data (e.g. X-rays) while maintaining the ability to predict diagnoses from the encoded data. The randomness in the encoder acts as the private key for the data owner. We quantify privacy as the number of attacker guesses required to re-identify a single image (guesswork). We propose a contrastive learning algorithm to estimate guesswork. We show empirically that differentially private methods, such as DP-Image, obtain privacy at a significant loss of utility. In contrast, Syfer achieves strong privacy while preserving utility. For example, X-ray classifiers built with DP-image, Syfer, and original data achieve average AUCs of 0.53, 0.78, and 0.86, respectively.