Don,
It's strictly based on what your governing documents say.
Although, VA Corporate law (specifically
VA ยง 13.1-854) allows an Association to adopt qualifiers (what qualifications an individual must meet to serve as Director), most governing documents (when initially written) are silent about qualifiers.
If your governing documents are silent (i.e. there are no qualifiers that say a Director must be in good standing, a member of the Association, etc.) then anyone (even Jon from NY or JohnB from SC or I) could serve on your Board if we were so inclined and properly elected or appointed.
The solution is to:
a) make sure that there is enough participation from the membership so everyone can choose whom they want to serve (and prevent vacancies which allow the Board to appoint someone to fill the vacancy).
b) gather support and recall the individual(s) or simply don't reelect them.
c) gather support so amendments to the Bylaws can be made to have specific qualifiers on who may serve.