Why do we calculate energy by integrate the Signal squared? What's the interesting thing in the square of a signal?
I know integrating gave us the sum of the differentiated energies, but why the Energy is the square of the signal?
 A: As pointed out in the comments, for signal processing (which is where we see the squares most frequently) Parseval's theorem tells us:

the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform

which is ultimately a statement about the unitary nature of Fourier transforms (as noted in the linked article)
Another way to understand this is to consider Ohm's law which relates voltage $v(t)$ to current $i(t)$ and resistance $R$:
$$v(t) = R i(t)$$
Power $p(t)$ is the product of current $i(t)$ and voltage $v(t)$:
$$p(t) = i(t)v(t)$$
If we use Ohm's law, we can make the substitution:
$$p(t) = Ri^2(t)$$
To find the energy, we integrate power over time:
$$E = R\int_{-\infty}^{\infty} i^2(t) dt$$
Which on a per ohm basis is:
$$E = \int_{-\infty}^{\infty} i^2(t) dt$$
If we want to generalize, and recognize that a signal can be complex, we can rewrite the equation as:
$$E = \int_{-\infty}^{\infty}  |x(t)|^2 dt$$ 
So you can see that the energy for practical purposes is derivable principally because of the relationship given by Ohm's law.  
A: 
What's the interesting thing in the square of a signal?

In signal processing and Fourier analysis, squares are interesting because they are simple to calculate and because they satisfy useful mathematical theorems, like the Parseval or the Plancherel theorem.
In physics, squares are interesting also because often (but not always) energy turns out to be quadratic function of some prominent measurable physical quantity. For example, energy stored in a capacitor is proportional to square of its voltage ($\frac{1}{2}CU^2$).

I know integrating gave us the sum of the differentiated energies,
  but why the Energy is the square of the signal?

By square of signal I assume you mean $\int_{t_1}^{t_2}S(t)^2 dt$ where $S(t)$ is the 'signal'. If that is so, it is not true that this square of signal always gives value of some physically sensible energy. Formally, if $S(t)$ is quadratically integrable function of time, one can always define 'energy' as the above square of the signal.
But whether that is some physically meaningful energy does not follow automatically from the equations for $S(t)$, but has to be derived from the underlying physics of the phenomena.
If $S(t)$ is macroscopic electric field component of linearly polarized wave in the direction of polarization, this is closely related to energy that such wave transports from time $t_1$ to time $t_2$ through fixed unit area of cross section. Physically, this is because the square of electric field in the Poynting theorem can be interpreted as EM energy density.
On the other hand, if $S(t)$ is temperature of some object as a function of time, the 'square of the signal' is a useless quantity, according to current knowledge of physics it has no useful relation to any physical energy of the object.
A: This is not a full answer, it is very partial but it gives the flavor of why we take the square of the signal, and why do we integrate over time. The origin of these things is in the FLUX of a signal. When on the path of a signal we place a detector, what impinges on it, is the flux (defined as quantity of incoming energy/unit surface in the unit time).
So, in order to get what amount of energy impinges on the detector in an interval of time, e.g. between t_1 and t_2, we have to integrate the flux between t_1 and t_2.
Now, let me take the example of an electromagnetic plane wave, mathematically it's the easiest example. The Poynting vector tells us its flux,
S = E x H,
where E is the electric field, H the magnetic field, and x indicates vector product.
Now the things becomes simple because for the plane wave the direction of the product E x H is along the direction of propagation of the wave, and H is proportional to E. 
This is why, up to some constant factor, we need E^2 - of course, for E dependent on t we need E(t)^2 - and why we integrate over this.
Now, why there appears the absolute square is because sometimes we work with E represented as a complex number.
I tried to give you an intuitive explanation, I hope it helps,
Sofia
A: I was looking for an answer to the same question as that asked here but I wans't fully satisfied with the given answers, so I kept looking and a good answer that I stumbled upon is the one given here: http://www.gaussianwaves.com/2013/12/power-and-energy-of-a-signal/.
Hope it will serve to others well as it served to me.
Cheers
