A USSC decision several years ago assured that students have the same First Amendment protection as adults, when a school attempted to stop students from wearing arm bands in protest of the Vietnam War.
In the case you're referring to, Tinker v. Des Moines, the Supreme Court ruled that school officials can prohibit student speech when they can reasonably forecast that the expression will create a substantial disruption or invade the rights of others. They ruled the arm bands did not.
But in Harper v. Poway Unified School District, the Ninth Circuit ruled that school officials could prohibit student Tyler Chase Harper from wearing shirts bearing messages such as “BE ASHAMED, OUR SCHOOL EMBRACED WHAT GOD HAS CONDEMNED” and “HOMOSEXUALITY IS SHAMEFUL.” Harper is permitted to wear that shirt on the street, however.
In 1988, the U.S. Supreme Court ruled in Hazelwood School District v. Kuhlmeier that public school officials can censor school-sponsored student expression as long as they have a valid educational reason for doing so.