Gauss's law: flux is invariant under deformation

What you are seeing: a point charge $q$ inside (or outside) a deformable closed 2D loop. The flux of the electric field through the loop is computed numerically by Simpson quadrature on $\oint \mathbf{E} \cdot \hat n \, ds$. For 2D Gauss the result is $q / \epsilon_0$ whenever the charge sits inside, and zero when it sits outside, no matter how the loop is deformed.

Slide the ellipse semi-axes or switch to the blob shape. The readout reports the computed flux and the ratio to the theoretical value $q / \epsilon_0$. Move the charge position out past the ellipse boundary and the flux drops to zero.