The case is Castine v. Zurlo, decided on June 23. Your first instinct is that the Town violated the Free Speech clause because plaintiff is being punished for running for office. That's true, but there is more to it. Under the Supreme Court's 1968 decision in Pickering v. Board of Education, the government can discipline and even punish public employees if their protected speech or activity would disrupt efficient government operations, and that potential for disruption outweighs the value of the plaintiff's speech. An example of this was a 2006 case where the Court of Appeals said the NYPD could discipline police officers who engaged in racist off-duty speech that could impair the performance of their duties (since the public has no confidence in racist police officers).
This case is not as dramatic as the racist cops case, but the plaintiff still loses. The Court of Appeals (Leval, Jacobs and Pooler) says, "first, it was reasonable for Defendants to expect that Castine’s candidacy for elective office would create a conflict of interest with her position as Election Commissioner and that it would impair the integrity of Clinton County elections to permit Castine to occupy those two roles simultaneously. Second, it is clear that Defendants temporarily suspended Castine in order to advance the permissible objective of enforcing local law and minimizing the potential damage to the integrity of local elections. Third, the Defendants’ interest in avoiding disruption to the work of the Election Commission outweighs Castine’s interest in running for public elective office while simultaneously serving as Election Commissioner."
In light of this evidence, the County Legislature did what it had to do. The Court explains:
The Clinton County legislature reasonably concluded that the Board’s ability to supervise elections fairly, as well as the public perception of its fairness in performing that function, would be impaired if the persons involved in exercising the supervisory responsibility were themselves candidates in the elections they were supervising. This would present an unacceptable conflict of interest. The legislature therefore enacted the law disqualifying candidates for elective office from simultaneous employment in the Board of Elections. At least the appearance of fairness, and in all likelihood the fact as well, would have been substantially impaired if a candidate for election were to participate in the functions of the Board of Elections supervising that election. The act of the Defendants in removing Castine from her position as Commissioner for the duration of her campaign for elective office reasonably gave effect to this altogether reasonable provision of law. Although without doubt the enforcement of the law caused some curtailment of Castine’s exercise of her First Amendment rights, the Pickering test authorizes such a result.