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In recent years, there has been a growing interest in the use of single-photon avalanche diode (SPAD) in optical wireless communication (OWC). SPAD operates in the Geiger mode and can act as a photon counting receiver obviating the need for a transimpedance amplifier (TIA). Although a SPAD receiver can provide higher sensitivity compared to the traditional linear photodetectors, it suffers from the dead-time-induced nonlinearity. To improve the data rates of SPAD-based OWC systems, optical orthogonal frequency division multiplexing (OFDM) can be employed. This paper provides a comprehensive theoretical analysis of the SPAD-based OWC systems using OFDM signalling considering the effects of signal clipping, SPAD nonlinearity, and signal-dependent shot noise. An equivalent additive Gaussian noise channel model is proposed to describe the performance of the SPAD-based OFDM system. The statistics of the proposed channel model and the analytical expressions of the signal-to-noise ratio (SNR) and bit error rate (BER) are derived in closed forms. By means of extensive numerical results, the impact of the unique receiver nonlinearity on the system performance is investigated. The results demonstrate new insights into different optical power regimes of reliable operation for SPAD-based OFDM systems even well beyond SPAD saturation level.