A cylinder consists of two disks, with opposite orientation, and a lateral side. By symmetry, the flux through the lateral side is zero. Since the two disks have opposite orientations, again by symmetry, the sum of the flux through them is also zero. This is consistent with what you would expect from Gauss' Law, since the cylinder contains zero charge.
However, what you have here is quite different from a cylinder, since a disk is not a Gaussian surface, meaning that you cannot use Gauss' Law*. Instead you need to calculate the electric field at each point on the disk using Coulomb's Law, and then integrate. If you think about which way the electric field points at each point on the disk, it should become intuitively clear why the flux is non-zero. The key thing that distinguishes this from the cylinder, is that here, you have a single disk, whereas the cylinder has two disks whose flux cancels out because of their opposite orientations.
*In principle, there is a way to indirectly use Gauss' Law by finding the solid angle of the cone with the disk as its base and apex at one of the point charges, and then finding the flux through that solid angle.