To understand electric flux, you have first to understand the concept of field lines or the fact that the electric field is a vector field or both :). Electric flux is simpy how many field lines pass through the surface (of which surface we want to calculate the electric flux through).
So electric flux depends not only on the electric field but also on what surface we choose to calculate the flux that is passing through that surface.
If the electric field is zero over the surface of interest then the flux through that surface is zero also.
However it can be the case that the total flux through a surface is zero but the electric field is not zero in all the points of the surface. To imagine how this can happen you have to think of a closed surface like the surface of a sphere. Imagine that you put this imaginary sphere inside a charged capacitor. The electric field lines will enter inside the sphere from one hemisphere (this creates a negative electric flux because the field lines are entering inside ) and will exit from the other hemisphere (this creates an equal and positive electric flux because the field lines are exiting). The total flux will be the sum of flux for one hemisphere and for the other, and will be zero.