Approximate computing emerged as a design alternative to boost design efficiency by leveraging the intrinsic error resiliency of many applications. Several error-resilient and compute-intensive applications such as signal, image, and video processing, computer vision, and supervised machine learning perform mean squared error (MSE) estimation during the runtime demanding dedicated squarer logic units in their hardware accelerators. This work introduces an approximate Radix-4 squarer unit architecture (AxRSU). Our AxRSU proposal reduces the encoder complexity and the number of required partial products, which considerably boosts energy and circuit area savings. We demonstrate the AxRSU error-quality tradeoff in an SSD (Sum Squared Difference) hardware accelerator as a case study targeting a video processing application. We offer a new Pareto front with eighth optimal AxRSU solutions ranging 52-97% of cross-correlation (i.e., precision) for savings of 15-47% in energy consumption and 12-32% in circuit area.