The short answer to this question is: Only owners can serve on the Board.  But what constitutes an owner?  In order to be a legal owner in a condominium association, the owner's name must appear on the warranty deed for the property. 

In the event the unit owner is a corporation, partnership, trust or other legal entity other than a natural person or persons, then any director or officer, beneficiary or manager may serve as a member of the Board.

Spouses and other family members who reside in a unit cannot serve on the Board unless their name appears on the deed to the property.  This prevents any individual who does not have a legal stake in the property from making decisions about its operation.

Unsure whether your Board members are, in fact, deeded owners?  You can search for this information on the Cook County Recorder of Deeds' website by owner name or Property Index Number (PIN).  The Warranty Deed document will list all owners who are on the deed for the property.