Is it possible to transmit square waves EM signal through air?
A square wave can be seen as a superposition of sine waves (of odd multiple frequencies of the lowest frequency). I.e. a 1MHz square wave is composed of 1MHz, 3MHz, 5MHz, 7MHz etc. sine waves.

So as long as everything behaves linearly and there is no dispersion (i.e. no frequency dependency of \$n\$; both conditions are satisfied in case of vacuum) theoretically it is possible.
Be aware, however, that it requires a (theoretically inifinite) wide bandwidth.