Long-range spin-spin interactions are known to generate non-equilibrium dynamics which can squeeze the collective spin of a quantum spin ensemble in a scalable manner, leading to states whose metrologically useful entanglement grows with system size. Here we show theoretically that scalable squeezing can be produced in 2d U(1)-symmetric systems even by short-range interactions, i.e. interactions that at equilibrium do not lead to long-range order at finite temperatures, but rather to an extended, Berezhinski-Kosterlitz-Thouless (BKT) critical phase. If the initial state is a coherent spin state in the easy plane of interactions, whose energy corresponds to a thermal state in the critical BKT phase, the non-equilibrium dynamics exhibits critical slowing down, corresponding to a power-law decay of the collective magnetization in time. This slow decay protects scalable squeezing, whose scaling reveals in turn the decay exponent of the magnetization. Our results open the path to realizing massive entangled states of potential metrological interest in many relevant platforms of quantum simulation and information processing -- such as Mott insulators of ultracold atoms, or superconducting circuits -- characterized by short-range interactions in planar geometries.