Metaverse-enabled digital healthcare systems are expected to exploit an unprecedented amount of personal health data, while ensuring that sensitive or private information of individuals are not disclosed. Machine learning and artificial intelligence (ML/AI) techniques can be widely utilized in metaverse healthcare systems, such as virtual clinics and intelligent consultations. In such scenarios, the key challenge is that data privacy laws might not allow virtual clinics to share their medical data with other parties. Moreover, clinical AI/ML models themselves carry extensive information about the medical datasets, such that private attributes can be easily inferred by malicious actors in the metaverse (if not rigorously privatized). In this paper, inspired by the idea of "incognito mode", which has recently been developed as a promising solution to safeguard metaverse users' privacy, we propose global differential privacy for the distributed metaverse healthcare systems. In our scheme, a randomized mechanism in the format of artificial "mix-up" noise is applied to the federated clinical ML/AI models before sharing with other peers. This way, we provide an adjustable level of distributed privacy against both the malicious actors and honest-but curious metaverse servers. Our evaluations on breast cancer Wisconsin dataset (BCWD) highlight the privacy-utility trade-off (PUT) in terms of diagnosis accuracy and loss function for different levels of privacy. We also compare our private scheme with the non-private centralized setup in terms of diagnosis accuracy.