Neutral-atom arrays trapped in optical potentials are a powerful platform for studying quantum physics, combining precise single-particle control and detection with a range of tunable entangling interactions. For example, these capabilities have been leveraged for state-of-the-art frequency metrology as well as microscopic studies of entangled many-particle states. Here we combine these applications to realize spin squeezing-a widely studied operation for producing metrologically useful entanglement-in an optical atomic clock based on a programmable array of interacting optical qubits. In this demonstration of Rydberg-mediated squeezing with a neutral-atom optical clock, we generate states that have almost four decibels of metrological gain. In addition, we perform a synchronous frequency comparison between independent squeezed states and observe a fractional-frequency stability of 1.087(1) × 10 at one-second averaging time, which is 1.94(1) decibels below the standard quantum limit and reaches a fractional precision at the 10 level during a half-hour measurement. We further leverage the programmable control afforded by optical tweezer arrays to apply local phase shifts to explore spin squeezing in measurements that operate beyond the relative coherence time with the optical local oscillator. The realization of this spin-squeezing protocol in a programmable atom-array clock will enable a wide range of quantum-information-inspired techniques for optimal phase estimation and Heisenberg-limited optical atomic clocks.