We study the classical and quantum fluctuations of a general damped forced oscillator close to a bifurcation instability. Near the instability point, the fluctuations are strongly phase correlated and are squeezed. In the limit of low damping, it is shown that the system has universal features when scaled with the damping. The same scaling law applies to the classical and to the quantum regimes. We furthermore show that the coupling to the environment is crucial in the generation of squeezed fluctuations.