A person told me the following:

I don't live in this country by choice.

What he meant was that he didn't select this country as the country of living. He, probably, became a refugee.

Does a native speaker of English use the words 'by choice' in this context? Of course, it is correct.
Would a native speaker of English use those words? Yes.
Yes, a negated verb combined with "by choice" is a very common way to say that you had no choice in the matter. Circumstances dictated the course of events.