Speech is known to carry health-related attributes, which has emerged as a novel venue for remote and long-term health monitoring. However, existing models are usually tailored for a specific type of disease, and have been shown to lack generalizability across datasets. Furthermore, concerns have been raised recently towards the leakage of speaker identity from health embeddings. To mitigate these limitations, we propose WavRx, a speech health diagnostics model that captures the respiration and articulation related dynamics from a universal speech representation. Our in-domain and cross-domain experiments on six pathological speech datasets demonstrate WavRx as a new state-of-the-art health diagnostic model. Furthermore, we show that the amount of speaker identity entailed in the WavRx health embeddings is significantly reduced without extra guidance during training. An in-depth analysis of the model was performed, thus providing physiological interpretation of its improved generalizability and privacy-preserving ability.