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.

No, perfect mathematical square waves don't exist in real world because square wave is not a continuous function (it does not have a derivative at the step). Therefore you can only approximate a square wave and the approximation does have very high frequencies, and at some point the antenna would not be able to send these so it would be a low-pass filter.

In a more general case compared to the answers above, nothing can be stoped or started in zero time ie instantly. To do so would imply an infinitely high frequency component which would translate to infinite energy. The constraining factors are speed of light limitation of Special Relativity and Quantum Mechanics Uncertainty Principle.

The sharper the transition you want, the more energy you have to pump into the system