Epsilon-near-zero and epsilon near-pole materials enable reflective systems supporting a class of symmetry-protected and accidental embedded eigenstates (EEs) characterized by a diverging phase resonance. Here we show that pairs of topologically protected scattering singularities necessarily emerge from EEs when a non-Hermitian parameter is introduced, lifting the degeneracy between oppositely charged singularities. The underlying topological charges are characterized by an integer winding number and appear as phase vortices of the complex reflection coefficient. By creating and annihilating them, we show that these singularities obey charge conservation, and provide versatile control of amplitude, phase, and polarization in reflection, with potential applications for polarization control and sensing.