It is possible to add a filter to the_content that takes the query_var (in this case ‘uuid=237237’) from the url and appends it to any outgoing links. But that is beside the point. Because if the link leads to a place outside the current WordPress install the query_var will be lost anyway, even if the user returns afterwards.
Using query_var as an authentication method is simply not a good idea, because it is not persistent and easily falsified. WP has a built in user registration system to let users access restricted information repeatedly. There are plenty of plugins that provide extended interfaces if needed.
The best course to follow here is not trying to adapt your plugin, but ditch it for one that enables proper user registration and access.