Squeezing is a nonclassical feature of quantum states that is a useful resource, for example, in quantum sensing of mechanical forces. Here, we show how to use optimal control theory to maximize squeezing in an optomechanical setup with two external drives and determine how fast the mechanical mode can be squeezed. For the autonomous drives considered here, we find the inverse cavity decay to lower-bound the protocol duration. At and above this limit, we identify a family of protocols leveraging a two-stage control strategy, where the mechanical mode is cooled before it is squeezed. Identification of the control strategy allows for two important insights: to determine the factors that limit squeezing and to simplify the time dependence of the external drives, making our protocol readily applicable in experiments.
