A: Non-members can certainly be members of a board as ex-officio members. For example, the Governor of the state is often an ex-officio member of the state university's board even though not a member of the university. Another example is when the CEO of the XYZ company is on the board of directors of the ABC company. (RONR p. 473.) But ex-officio members are not elected; they are members by virtue of their office.

I find nothing in RONR that prevents a non-member from being on the board, but you must check your association's bylaws to see what they say.