We obtain a lower bound on the sum of two orthogonal spin component variances in a plane. This gives a planar uncertainty relation which holds even when the Heisenberg relation is not useful. We investigate the asymptotic, large-J limit and derive the properties of the planar quantum squeezed states that saturate this uncertainty relation. These states extend the concept of spin squeezing to any two conjugate spin directions. We show that planar quantum squeezing can be achieved experimentally as the ground state of a Bose-Einstein condensate in two coupled potential wells with a critical attractive interaction. These states reduce interferometric phase noise at all phase angles simultaneously. This is useful for one-shot interferometric phase measurements where the measured phase is completely unknown. Our results can also be used to derive entanglement criteria for multiple spins J at separated sites, with applications in quantum information.