The armband could be considered symbolic speech, which is protected under the First Amendment.
Wearing an armband definitely constitutes symbolic speech. That point alone, however, does not determine the outcome of the case.
The First and Fourteenth Amendments provide that the government is not permitted to censor the content of speech. However, neither amendment recognizes a right to engage in speech wherever one pleases. The Constitution does not guarantee the right to speak in a government building, just as it does not guarantee the right to bear arms in a government building.
There is no doubt in my mind that the government is fully entitled to restrict speech on its own property. The regulation in question in
Tinker is completely constitutional.