Physical health records belong to healthcare providers, but the information contained within belongs to each patient. In an increasing manner, more health-related data is being acquired by wearables and other IoT devices following the ever-increasing trend of the "Quantified Self". Even though data protection regulations (e.g., GDPR) encourage the usage of privacy-preserving processing techniques, most of the current IoT infrastructure was not originally conceived for such purposes. One of the most used communication protocols, MQTT, is a lightweight publish-subscribe protocol commonly used in the Edge and IoT applications. In MQTT, the broker must process data on clear text, hence exposing a large attack surface for a malicious agent to steal/tamper with this health-related data. In this paper, we introduce MQT-TZ, a secure MQTT broker leveraging Arm TrustZone, a popular Trusted Execution Environment (TEE). We define a mutual TLS-based handshake and a two-layer encryption for end-to-end security using the TEE as a trusted proxy. We provide quantitative evaluation of our open-source PoC on streaming ECGs in real time and highlight the trade-offs.