We demonstrate that data-driven system identification techniques can provide a basis for effective, non-intrusive model order reduction (MOR) for common circuits that are key building blocks in microelectronics. Our approach is motivated by the practical operation of these circuits and utilizes a canonical Hammerstein architecture. To demonstrate the approach we develop a parsimonious Hammerstein model for a nonlinear CMOS differential amplifier. We train this model on a combination of direct current (DC) and transient Spice (Xyce) circuit simulation data using a novel sequential strategy to identify the static nonlinear and linear dynamical parts of the model. Simulation results show that the Hammerstein model is an effective surrogate for the differential amplifier circuit that accurately and efficiently reproduces its behavior over a wide range of operating points and input frequencies.