Can I use OneNote with Owncloud or Nextcloud?

No. OneNote absolutely needs COBALT as sync protocol which is only provided by OneDrive/OneDrive for Business and Sharepoint. A file-based synchronization of notebooks (like Dropbox or Google Drive do) leads to lots of traffic and massive sync problems and the WEBDAV protocol (which is needed by Owncloud or Nextcloud) is not supported by OneNote.