Hyperlinks change when I change servers

This is a general question, so I will give a general answer: you can't. How can you expect Excel to know where you moved the source file unless you somehow tell it?

The first question I have is, "What do you mean by server and 'moving' your laptop to it?"

If by "server" you mean a mapped drive, then the solution is to use the full pathname rather than the drive letter. So instead of something like J:/Data you may have to use something like //server name/project/Data.