Do navigation databases (by e.g. Aeronavdata or Jeppesen) include company routes?

The company routes are maintained by your company. Some FMS installations have a computer program where you can create the company routes and upload them to each FMS. Other FMS requires each FMS to be programmed.

All FMS have the ability to see if the routes have been changed due to a database change. If the pilots wanted to use those company routes the FMS would require them to verify the routes are stored correctly.