Growing concerns over data privacy underscore the need for deep learning methods capable of processing sensitive information without compromising confidentiality. Among privacy-enhancing technologies, Homomorphic Encryption (HE) stands out by offering post-quantum cryptographic security and end-to-end data protection, safeguarding data even during computation. Prior research on encrypted training has primarily focused on logistic regression, model fine-tuning, or relied on multi-party computation. This is largely due to the substantial computational overhead and algorithmic complexity involved in training deep Neural Networks (NNs) under HE. In this paper, we present ReBoot, the first framework to enable fully encrypted and non-interactive training of Multi-Layer Perceptrons (MLPs) using CKKS bootstrapping. ReBoot introduces a novel HE-compliant NN architecture based on local error signals, specifically designed to minimize multiplicative depth and reduce noise accumulation during training. It employs a tailored packing strategy that leverages real-number arithmetic through CKKS \textit{SIMD} operations, significantly lowering both computational and memory overhead. We evaluate ReBoot on both image and tabular benchmarks, demonstrating up to $+6.83\%$ improvement in test accuracy over existing solutions, while reducing training latency by up to $8.83\times$. ReBoot is made available to the scientific community as a public repository.