With advances seen in deep learning, voice-based applications are burgeoning, ranging from personal assistants, affective computing, to remote disease diagnostics. As the voice contains both linguistic and para-linguistic information (e.g., vocal pitch, intonation, speech rate, loudness), there is growing interest in voice anonymization to preserve speaker privacy and identity. Voice privacy challenges have emerged over the last few years and focus has been placed on removing speaker identity while keeping linguistic content intact. For affective computing and disease monitoring applications, however, the para-linguistic content may be more critical. Unfortunately, the effects that anonymization may have on these systems are still largely unknown. In this paper, we fill this gap and focus on one particular health monitoring application: speech-based COVID-19 diagnosis. We test three anonymization methods and their impact on five different state-of-the-art COVID-19 diagnostic systems using three public datasets. We validate the effectiveness of the anonymization methods, compare their computational complexity, and quantify the impact across different testing scenarios for both within- and across-dataset conditions. Additionally, we provided a comprehensive evaluation of the importance of different speech aspects for diagnostics and showed how they are affected by different types of anonymizers. Lastly, we show the benefits of using anonymized external data as a data augmentation tool to help recover some of the COVID-19 diagnostic accuracy loss seen with anonymization.