To explore the benefits of approximate computing, this article proposes an approximate partial product generator for squarer (APPGS). Using APPGS, three designs of approximate radix-4 Booth squarers (ABS1, ABS2, and ABS3) are proposed. APPGS produces approximate partial products in $r$ number of least significant columns of the partial product matrix. ABS2 and ABS3 utilize approximate adders and compressors with a novel input signal rearrangement method for the accumulation of approximate partial products. Moreover, the ABS3 features an error recovery module at $k$ number of most significant columns of the approximate partial products. The proposed squarers with different values of $r$ and $k$ are simulated using 45-nm CMOS technology. The results indicate that the proposed squarers achieve optimized performance for both hardware and accuracy metrics. Compared to the exact Booth squarer, the 16-bit ABS1 with $r=16$ achieves a reduction of 13.6%, 22.2%, and 13.7% in power, delay, and area, respectively, with a normalized mean error distance (NMED) of $4.6\times 10^{-6}$ . The ABS2 has power, delay, and area savings of 25.8%, 33.8%, and 19.8%, respectively, with an NMED of $7.2\times 10^{-6}$ . The ABS3 with $k=6$ has 18.5% reduction in power, 29.4% reduction in delay, and 16.9% reduction in area with an NMED of $0.56\times 10^{-6}$ . The performance of the proposed squarers is evaluated with a telecommunication application, where the ABS3 with $k=6$ produces an output signal with a signal-to-noise ratio of 32.45 dB.