Neb. Rev. Stat. §§ 23-3534
The elective officers of a local hospital district shall be a board of directors consisting of five members. The members of the first board shall be appointed by the county board and shall be so appointed that two members shall serve terms ending on the first Tuesday in June following the first statewide primary election following the initial appointment, and three shall serve terms ending on the first Tuesday in June following the second statewide primary election following the initial appointment.
Members shall be elected as provided in section 32-550. All registered voters of this state who reside within the hospital district on or before the day of the election shall be entitled to vote in such hospital district election.
Any vacancy upon such board occurring other than by the expiration of a term shall be filled by appointment by the remaining members of the board of directors. Any person appointed to fill such vacancy shall serve for the remainder of the unexpired term. If there are vacancies in the offices of a majority of the members of the board, there shall be a special election conducted by the Secretary of State to fill such vacancies.
Neb. Rev. Stat. §§ 23-3534