An electronic device may have a housing in which components such as a display are mounted. A strain gauge may be mounted on a layer of the display such as a cover layer or may be mounted on a portion of the housing or other support structure. The layer of material on which the strain gauge is mounted may be configured to flex in response to pressure applied by a finger of a user. The strain gauge may serve as a button for the electronic device or may form part of other input circuitry. A differential amplifier and analog-to-digital converter circuit may be used to gather and process strain gauge signals. The strain gauge may be formed form variable resistor structures that make up part of a bridge circuit that is coupled to the differential amplifier. The bridge circuit may be configured to reduce the impact of capacitively coupled noise.