Why is the energy density of the electric field produced by two unlike charges positive, even though their potential energy is negative? Consider two stationary charges, one positive the other negative.  Their potential energy is clearly negative.  So you would expect that the energy density of the associated electric field would also be negative.  But it isn’t.  It’s the square of the electric field and therefore positive.  Why isn’t the energy density negative like the potential energy?
 A: The electric field energy is indeed positive for the case of two opposite charges, but it is smaller than the electric field energy when the charges are separated. The difference is the potential energy of interaction of the two charges, which includes the self-energy, the field energy for a single charge, inside the mass of the charge.
You can see that the energy is decreasing when you bring to equal charges together, because when they are right on top of each other, the field is zero. To see mathematically that the decrease is the potential energy, consider the energy integral
$$ \int |\nabla \phi|^2 dV = - \int \phi \nabla^2 \phi = \int \phi \rho $$
Where the first equality is an integration by parts, and $\rho$ is the charge density. When you can write the field $\phi$ as a sum of two contributions from two separate small charged spheres, the pontential energy of the mutual interaction is given by the charge of one times the potential generated by the other. There is also the self-pontential, which is the energy of the charge in its own field, which is classically divergent for a point charge, and can be neglected because you assume the charge radius is unchanged as the particles move around, so that this energy is constant.
A: The field generated by the two charges $q_1$ and $q_2$ is the sum of the field generated by the first and the second charge, let's say  $ \vec E_1+\vec E_2$. When you square this sum and integrate in the whole space you got
three terms. Integrating all over the space the square of $ \|\vec E_1\|^2$ and $\|\vec E_2\|^2$ we obtain the self energy of the two charges on their own i.e. the energy of the field of a lone charge. Each of them is positive and  the sum of these two is bigger than the integral of the  double scalar product of $ 2(\vec E_1\cdot\vec E_2)$ in the whole space. This last is negative and is the usual energy obtained multiplying one charge  for the potential generated by the other. 
Nevertheless a  question troubles me: why this is not in standard textbooks?  Furthermore, not having a sound background in Physics, has somebody any idea of what is the physical meaning of this   $ \|\vec E\|^2$ self energy. How does it relates with the mass of the electron and how this energy relates with the photons associated with the electromagnetic wave there should be every time a positron-electron couple rise.   
A: Potential energy of interaction is about interaction and can be negative.
The electric field created by two charges for a third one is a vector sum of two fields:
$\vec{E}=\vec{E}_1(\vec{r}_3-\vec{r}_1)+\vec{E}_2(\vec{r}_3-\vec{r}_2)$. 
It defines the corresponding force. From it you can derive the potential energy of interaction of the third charge with two original ones. Squared, it does not represent anything about interaction of the first two.
A: Here, you should refer to the definition of energy density, the electric field energy density is defined as following:
$u = \frac{1}{2}\epsilon |\vec{E}|^2$
where $\epsilon$ is the permittivity of media.
therefore this energy density should always be positive.
The derivation of the electric energy density can be referred to the wiki post
btw, I think you made some mistake statement in your question on your example

"Their potential energy is clearly negative. "

We can look at you example of two charge particles, we can easily write out the potential everywhere in your case as
$\Phi(\vec{x}) = \frac{Q_1}{4\pi\epsilon |\vec{x}-\vec{r_1}|}+\frac{Q_2}{4\pi\epsilon |\vec{x}-\vec{r_2}|}$
where $\vec{r_1}$ and $\vec{r_2}$ are the position of these charges in space, if we assume $Q_1$ is positive, $Q_2$ is negative, then we can calculate the electric potential everywhere. Clearly, you'll notice that there is zero-potential surface perpendicular to the line connects these two charges, on the side of $Q_1$, potential is always positive, and on the side of $Q_2$, potential is always negative, then the electric potential energy at these two charges are both positive ($U_1 = \Phi_1Q_1 >0$, since $\Phi_1>0$, similarly, $U_2 = \Phi_2Q_2 >0$, cause $\Phi_2 <0$ and $Q_2<0$). So the electric potential energy is positive. Also you should note that this energy is only a part of total electric energy. 
