Recently developed single-photon avalanche diode (SPAD) array cameras provide single-photon sensitivity and picosecond-scale time gating for time-of-flight measurements, with applications in LIDAR and fluorescence lifetime imaging. As compared to standard image sensors, SPAD arrays typically return binary intensity measurements with photon time-of-arrival information from fewer pixels. Here, we study the feasibility of implementing Fourier ptychography (FP), a synthetic aperture imaging technique, with SPAD array cameras to reconstruct an image with higher resolution and larger dynamic range from acquired binary intensity measurements. Toward achieving this goal, we present (1) an improved FP reconstruction algorithm that accounts for discretization and limited bit depth of the detected light intensity by image sensors, and (2) an illumination angle-dependent source brightness adaptation strategy, which is sample-specific. Together, these provide a high-quality amplitude and phase object reconstruction, not only from binary SPAD array intensity measurements, but also from alternative low-dynamic-range images, as demonstrated by our simulations and proof-of-concept experiments.