What are the rights of a U.S. citizen concerning the time spending outside the U. S.

I have no idea what you mean by "rights." As a US citizen you can go anywhere another Country will allow you to go.

What exactly is your situation?

I have no idea what you mean by "rights." As a US citizen you can go anywhere another Country will allow you to go.

What exactly is your situation? I would like to know how long can you be living outside the U.S. without losing your U.S. Citizenship.

Forever.