Squeezed light, displaying less fluctuation than vacuum in some observable, is key in the flourishing field of quantum technologies. Optical or microwave cavities containing a Kerr nonlinearity are known to potentially yield large levels of squeezing, which have been recently observed in optomechanics and nonlinear superconducting circuit platforms. Such Kerr-cavity squeezing however suffers from two fundamental drawbacks. First, optimal squeezing requires working close to turning points of a bistable cycle, which are highly unstable against noise thus rendering optimal squeezing inaccessible. Second, the light field has a macroscopic coherent component corresponding to the pump, making it less versatile than the so-called squeezed vacuum, characterised by a null mean field. Here we prove analytically and numerically that the bichromatic pumping of optomechanical and superconducting circuit cavities removes both limitations. This finding should boost the development of a new generation of robust vacuum squeezers in the microwave and optical domains with current technology.