The corporation shall have a board of directors in which all powers of the corporation shall be vested and which shall consist of any number of directors, not less than three, all of whom shall be duly qualified electors of and taxpayers in the county. The directors shall serve as such without compensation except that they shall be reimbursed for their actual expenses incurred in and about the performance of their duties under this article. No more than one director shall be an officer or employee of the county, and in no event shall a judge of probate serve as such director. The directors shall be elected by the governing body of the county and they shall be so elected that they shall hold office for staggered terms. At the time of the election of the first board of directors, the governing body of the county shall divide the directors into three groups containing as nearly equal whole numbers as may be possible. The first term of the directors included in the first group shall be two years, the first term of the directors included in the second group shall be four years, the first term of the directors included in the third group shall be six years and thereafter the terms of all directors shall be six years; provided, that if at the expiration of any term of office of any director a successor thereto shall not have been elected, then the director whose term of office shall have expired shall continue to hold office until his successor shall be so elected. Any meeting held by the board of directors for any purpose whatsoever shall be open to the public.