The Hosts file is a file that provides cross references between names and IP addresses on a private network. The names and IP addresses of the three PC's on your network may be in the Hosts file (although this is not required as there are other ways for PCs to find each other on a network).
The Hosts file is a simple text file, and I assume from what you say that there is some physical connection (the hub) inside your building that has to be 'found' for your healthcare program to work.
If you connect to the database over the internet, then a Hosts file should not be required as translation between name and IP address is done by DNS servers on the Internet.
As your Hosts file is being deleted it may be that your anti-virus/firewall program has wrongly identified it as being infected and is deleting it.
If you have a clean copy of the required Hosts file from a safe source or you create it yourself with Notepad, then tell your Anti-virus program that this Hosts file is safe and is not infected. There is usually an exception list that lets you identify safe files.
Have you talked to the IT people responsible for this healthcare system.