The answer, like so many things in law, is that "it depends." Was the ring ever discussed when the engagement was broken off? How long has it been since then?
Most courts tend to take the view that an engagement ring is a conditional gift, so that if the marriage doesn't take place, the ring ought to be returned. However, if the ring was kept after some discussion, or it wasn't mentioned for many years, then this could change the situation.
There is no "statute of limitation" on this. That phrase refers to the time limit given to the State to issue criminal charges against someone, or for a private party to file a lawsuit.
I think that's what this person means by "statute of limitations" - the time limit for a person to file a private lawsuit. In this case, the lawsuit would possibly be over an alleged breach of contract. ORS 12.080 says that the statute of limitations for many cases over contract and/or taking of personal property is limited to 6 years.
You should definitely take your case to a lawyer and get a specific analysis.