For the past two years, high schools in the town of Enfield, Conn. have preferred the indoor comforts of a church for their graduation ceremonies. But lawyers are now moving to stop the practice after it left some students feeling increasingly out in the cold.

A federal suit filed in May alleged that the public schools had violated the U.S. Constitution, which restricts government entities such as public schools from endorsing a particular religion, the latest flash point in a growing national debate on whether graduations...