This paper addresses the design of multi-antenna precoding strategies, considering hardware limitations such as low-resolution digital-to-analog converters (DACs), which necessitate the quantization of transmitted signals. The typical approach starts with optimizing a precoder, followed by a quantization step to meet hardware requirements. This study analyzes the performance of a quantization scheme applied to the box-constrained regularized zero-forcing (RZF) precoder in the asymptotic regime, where the number of antennas and users grows proportionally. The box constraint, initially designed to cope with low-dynamic range amplifiers, is used here to control quantization noise rather than for amplifier compatibility. A significant challenge in analyzing the quantized precoder is that the input to the quantization operation does not follow a Gaussian distribution, making traditional methods such as Bussgang's decomposition unsuitable. To overcome this, the paper extends the Gordon's inequality and introduces a novel Gaussian Min-Max Theorem to model the distribution of the channel-distorted precoded signal. The analysis derives the tight lower bound for the signal-to-distortion-plus-noise ratio (SDNR) and the bit error rate (BER), showing that optimal tuning of the amplitude constraint improves performance.