In this work, we introduce EQ-Net: the first holistic framework that solves both the tasks of log-likelihood ratio (LLR) estimation and quantization using a data-driven method. We motivate our approach with theoretical insights on two practical estimation algorithms at the ends of the complexity spectrum and reveal a connection between the complexity of an algorithm and the information bottleneck method: simpler algorithms admit smaller bottlenecks when representing their solution. This motivates us to propose a two-stage algorithm that uses LLR compression as a pretext task for estimation and is focused on low-latency, high-performance implementations via deep neural networks. We carry out extensive experimental evaluation and demonstrate that our single architecture achieves state-of-the-art results on both tasks when compared to previous methods, with gains in quantization efficiency as high as $20\%$ and reduced estimation latency by up to $60\%$ when measured on general purpose and graphical processing units (GPU). In particular, our approach reduces the GPU inference latency by more than two times in several multiple-input multiple-output (MIMO) configurations. Finally, we demonstrate that our scheme is robust to distributional shifts and retains a significant part of its performance when evaluated on 5G channel models, as well as channel estimation errors.