Most modern health care systems I know of use normal database formats like Oracle or SQL Server. In ages past, some used flat files or a proprietary storage mechanism, but I think these are all but gone by now. With the privacy and security requirements surrounding health care data and the size of the data sets we're talking about in a modern system, it is nearly a requirement to use a standard database format.

That said, nearly any DBMS can work, but some will be better than others. If you're looking at a third party application, I'm sure their specifications or requirements document contains the information.