How will study as a non-degree student affect my graduate admissions chances later?
Will the non-degree option help my chances at being accepted to university A's MS program?

No.

Will the non-degree option hurt my chances at being accepted to university B's PhD program?

No.

Will they need to know about the non-degree if I'm eventually accepted to the degree option?

Do well in the course and then you'll be proud to share the transcript with them.