Why does Hilbert filter distorts the shape of the signal?
A Hilbert transform is dispersive, as in any constant phase shift over frequency, which will distort a signal that has bandwidth. This is because a fixed DELAY, which would have no distortion, has linear phase versus frequency. If you fix the phase versus frequency to be constant (as we do in a Hilbert transform), the delay at each frequency will be different and thus will cause significant distortion to the signal. 

This is demonstrated with my slide below that is used for explaining why linear phase filters are important specifically, but also helps demonstrate what a constant phase shift versus frequency would do. Note that delay specifically is the derivative of phase versus frequency:

$$delay = \frac{d\phi}{d\omega}$$

In the slide below, the first three Fourier components of a square wave signal are shown, along with their addition when they all add up in time (which would happen with a linear phase filter as the delay in time given the formula above would be the same for each frequency). The "Non-linear Phase" case shows what happens if the Fourier components are summed after going through different time delays, which occurs in non-linear phase filters, AND occurs if the composite signal goes through a system where the phase is constant for all frequencies (as the time delay for each frequency must be different in order for the phase to be the same).