As an emerging non-volatile memory (NVM) technology, spin-torque transfer magnetic random access memory (STT-MRAM) has received great attention in recent years since it combines the features of low switching energy, fast write/read speed, and high scalability. However, process variation and thermal fluctuation severely affect the data integrity of STT-MRAM, resulting in both write errors and read errors. Therefore, effective error correction codes (ECCs) are necessary for correcting memory cell errors. Meanwhile, the design of channel quantizer plays a critical role in supporting error correction coding for STT-MRAM. In this work, we propose a union bound analysis which can accurately predict the word error rates (WERs) of ECCs with maximum-likelihood (ML) decoding over the quantized STT-MRAM channel. The derived bound provides a theoretical tool for comparing the performance of ECCs with different quantization schemes at very low error rate levels without resorting to lengthy computer simulations. Moreover, we also propose a new criterion to design the channel quantizer by minimizing the WERs of ECC decoding that are obtained from the union bound analysis. Numerical results show that the proposed union-bound-optimized (UBO) quantizer can achieve better error rate performance than the state-of-art quantizers for STT-MRAM.