Convergence of physical and digital identity and integration of various individual records, such as patient data, into a united repository remains a serious challenge. On one hand, collecting relevant data can help clinicians, specialists and healthcare service providers to facilitate care for patients. On the other hand, Self-Sovereign identity and the right to control personal data comes into question, because patients do not handle their data explicitly. Distributed Ledger Technology (DLT) is a novel method which would allow to securely record time-stamped data and enable patient-driven health and identity records. In this paper, we review the state-of-the-art in Blockchain (BC)-based self-sovereignty and patient data records in healthcare. Our motivation is to investigate the potential of BC technology for use in the patient data and identity management. As a distributed decentralized technology, BC can be very beneficial, giving patients control over their own data and self-sovereign identity. To the extent of our knowledge, there is no literature covering the same concerns. More specifically, the focus is on solutions that aim the realization of holistic BC-based Electronic Health Records (EHR) and Patient Health Records (PHR). EHR and PHR are used to record patient data, such as the doctor's notes upon a visit and radiology images. Hence, they include critical information regarding patient's privacy and identity. Therefore, development of pure decentralized Healthcare Information Systems (HIS) is a great challenge in terms of architectural and technical structure of the systems. Designing robust and reliable EHR and PHR, which represent the foundation of many other healthcare services, relies on carefully finding the balance in a trade-off between many factors, such as level of decentralization, privacy, scalability and data throughput. In this paper, we review the state-of-the-art and provide an analysis on the design trade-offs.