Wireless communications systems beyond 5G are expected to utilize large available bandwidths at frequencies above 100 GHz in order to achieve data rates above 100 Gbit/s. However, the power consumption of the analog-to-digital converters (ADCs) for such systems is becoming a major challenge. Trading a reduced amplitude resolution for an increased temporal resolution by employing temporal oversampling w.r.t. the Nyquist rate is a possible solution to this problem. In this work, we consider a wireless communications system employing zero-crossing modulation (ZXM) and 1-bit quantization in combination with temporal oversampling at the receiver, where ZXM is implemented by combining runlength-limited (RLL) transmit sequences with faster-than-Nyquist (FTN) signaling. We compare the performance and complexity of four different soft-output equalization algorithms, namely, two approximations of the linear minimum mean squared error (LMMSE) equalizer, a BCJR equalizer and a deep-learning based equalizer, for such systems. We consider the mutual information (MI) between the input bits of the RLL encoder and the output log-likelihood ratios (LLRs) of the RLL decoder as a performance measure and evaluate it numerically. Our results demonstrate that one of the proposed LMMSE equalizers outperforms the competing algorithms in the low and mid signal-to-noise ratio (SNR) range, despite having the lowest implementational complexity.