It has nothing at all to do with being paid or not. “Free” here refers to free software a.k.a. libre, or free-as-in-freedom. In the case of “non-free network services” it really just means whether the app is tied to a particular service or whether it can be used with a different instance or self-hosted.
FastHub for example has the NonFreeNet anti-feature because it exclusively works with GitHub, which is proprietary and cannot be self-hosted, even though GitHub itself is gratis (does not cost money to use). However, LabCoat which is a GitLab client is not NonFreeNet, because although it does work with the hosted GitLab.com service, GitLab is free software and can be self-hosted, and LabCoat can be configured to use another instance instead of GitLab.com.