“Restricting their freedom” is a point of view, and the way it will make people want more the European Union is more determined by how the project will be born by the politicians and medias, and how the services will be organized.
Why would the European Civic or military service would be a particular and intolerable restriction of liberty whereas it already exists in many of the democratic European countries? Switzerland, Denmark, Finland, Austria, Hungary, Greece and Estonia still have this mandatory service. In France, it is only suspended (since 1997) and could be implemented by simple decree. The current geopolitical situation of Europe could conduce to a return to this service to the rest of Europe.
Then, the military service has contributed to the formation of the French nation in XIXth century, by giving to many different people a standardize language and way of life. Therefore, it could help to create a European nation, conscious of itself.
Finally, they would be paid, learn more about languages and cultures.