Do square wave exist?
As you know (since you mentioned the Fourier transform), a square wave can be represented (well, almost -- see below) as the sum of an infinite series of sine waves. But it would not be possible to send a true square wave through any real physical antenna: As you move along the infinite series, the frequencies get higher and higher, and eventually you'll reach frequencies your antenna can't transmit, for various reasons. If you look at a chart of the electromagnetic spectrum, you will find that radio waves above a certain frequency are called "light", and your antenna probably can't reach those frequencies no matter how good it is.

(But, indeed, if you have an antenna that is capable of transmitting over a wide bandwidth -- that is, from very low to very high frequencies -- and you send some approximation of a square wave over it, you will see very high frequencies appear, just as predicted by the Fourier transform.)

There's also another problem: You can't quite actually approach a true square wave shape from any finite sum of sine waves, no matter how many. This problem is much more theoretical, and unlikely to actually come up in practice, but it's called the Gibbs phenomenon. It turns out that no matter how high in frequency you go, your approximation of a square wave will always overshoot at the big jumps from low to high and high to low. The overshoot will get shorter and shorter in time, the better your approximation (the higher in frequency you go.) But it will never go down in magnitude; it converges to about 9% of the size of the jump.