Nanopore sequencing is a widely-used high-throughput genome sequencing technology that can sequence long fragments of a genome. Nanopore sequencing generates noisy electrical signals that need to be converted into a standard string of DNA nucleotide bases (i.e., A, C, G, T) using a computational step called basecalling. The accuracy and speed of basecalling have critical implications for every subsequent step in genome analysis. Currently, basecallers are mainly based on deep learning techniques to provide high sequencing accuracy without considering the compute demands of such tools. We observe that state-of-the-art basecallers (i.e., Guppy, Bonito) are slow, inefficient, and memory-hungry as researchers have adapted deep learning models from other domains without specialization to the basecalling purpose. Our goal is to make basecalling highly efficient and fast by building the first framework for specializing and optimizing machine learning-based basecaller. We introduce RUBICON, a framework to develop hardware-optimized basecallers. RUBICON consists of two novel machine-learning techniques that are specifically designed for basecalling. First, we introduce the quantization-aware basecalling neural architecture search (QABAS) framework to specialize the basecalling neural network architecture for a given hardware acceleration platform while jointly exploring and finding the best bit-width precision for each neural network layer. Second, we develop SkipClip, the first technique to remove all the skip connections present in modern basecallers to greatly reduce resource and storage requirements without any loss in basecalling accuracy. We demonstrate the benefits of QABAS and SkipClip by developing RUBICALL, the first hardware-optimized basecaller that performs fast and accurate basecalling. Our experimental results on state-of-the-art computing systems show that RUBICALL is a fast, accurate and hardware-friendly, mixed-precision basecaller. Compared to a highly-accurate state-of-the-art basecaller, RUBICALL provides a 16.56x speedup without losing accuracy, while also achieving a 6.88x and 2.94x reduction in neural network model size and the number of parameters, respectively. Compared to the fastest state-of-the-art basecaller, RUBICALL provides a 3.19x speedup with 2.97% higher accuracy. We show that QABAS and SkipClip can help researchers develop hardware-optimized basecallers that are superior to expert-designed models.