Strictly speaking, separation of church and state is a good deal more complicated than would seem given modern discussions.
Way back before the Civil War, it applied to the federal government being prohibited from establishing a national religion, hence it's often being referred to as the establishment clause.
The states were understood to have varying degrees of religious leanings: Maryland was established as a haven for Roman Catholics of English heritage; Pennsylvania still goes by the moniker Quaker State owing to its founding; Utah is famously linked to the Church of Latter Day Saints. The list goes on.
After the Civil War, and in a time when religious tolerance in the United States had progressed well beyond what had been typical in the 1600s, the Constitution was amended to specifically declare that individual rights the federal government is obliged to recognize must likewise be recognized by the state governments.
Thus, the freedom from a government-preferred religion, guaranteed by the First Amendment, is extended to preclude the states (or counties or towns) from establishing a religion preferred by a local government.
Of course, between Maryland's founding in 1634 as a colonial haven for Catholics and the passage of the Civil War Amendments more than two centuries later, plenty of religion had found its way into our traditions and laws.
Fast forward to the Bel Air Town Commissioners of our time, who have pondered the issue of whether the tradition of closing town offices on Good Friday should be continued.
Neither the state nor the county governments recognize Good Friday as a holiday (it's worth noting that the word holiday itself derives from the words holy day).
Still, just about the whole country shuts down for Christmas, an inherently religious holiday (not to mention holy day).
Also there's the matter of the tradition of U.S. currency bearing the words "In God We Trust."
Is closing a government office on a religious holiday (or on Saturdays and Sundays when Jews and most Christians, respectively, observe the Sabbath) the same thing as establishing a state religion? Or is it a gesture of respect to a substantial segment of society that holds a particular set of religious beliefs?
Such issues are best considered on a case by case basis. It does no one any harm to close public facilities on Rosh Hashana, Good Friday, Christmas or Eid (the Islamic festival of sacrifice), and it probably does a lot in the realm of advancing understanding among our many faiths (and even between the faithful and atheists).
Of course, there is another major consideration, that being just about every day on the calendar can be linked to one or more religious observances or days of national import (like the Fourth of July). The government can't close down for all of them, so it needs to choose carefully the ones it decides to close down for.
If closing on Good Friday is a tradition that's deemed important in town, there's nothing wrong with closing the offices, but if it makes more sense to synchronize the town's days off with other local governments, that's fine too, just so long as no one is required to attend a worship service if they choose not to.