Motivated by recent surprising experimental results for the noise output of superconducting microfabricated resonators used in quantum computing applications and astronomy, we develop a fully quantum theoretical model to describe quantum dynamics of these circuits. Building on theoretical techniques from quantum optics, we calculate the noise in the output voltage that results in the presence of two-level systems created by an amorphous dielectric film. The theory predicts squeezing for the noise in the amplitude quadrature with respect to the input noise, which qualitatively reproduces the noise ellipse observed in experiment. We show that noise enhancement along the phase direction persists at partially saturating fields and for pump frequencies away from resonance, as found in experiments. Our results show that observable squeezing is an expected consequence of two-level systems in strongly driven superconducting resonators.