Two plastic sheets with charged densities as shown:
I'm trying to find the field at $B$. I obtained the correct answer by adding up the fields created by each charge density. But I realized that since the field is uniform in the region between the two sheets, I should be able to make a Gaussian surface between the sheets with the shape of a box and with one edge at $B$. Thus, the flux would be
$$\int{E \cdot dA} ~=~ EA ~=~ \frac{q_{encl}}{\epsilon_{0}}~=~0.$$
Because there would be no charged enclosed inside the surface. However, that means that $EA$ is $0$ (Note the integral reduced to $EA$ because $E$ is uniform). Since $A$ is not zero (it is the area of two sides of the box), this means that $E$ must be zero. However, $E$ is not zero there, as you can see by adding $\frac{\sigma}{2\epsilon_0}$ for each charge density.
What am I doing wrong when using Gauss' law?