Precise monitoring of respiratory rate in premature infants is essential to initiate medical interventions as required. Wired technologies can be invasive and obtrusive to the patients. We propose a Deep Learning enabled wearable monitoring system for premature newborn infants, where respiratory cessation is predicted using signals that are collected wirelessly from a non-invasive wearable Bellypatch put on infant's body. We propose a five-stage design pipeline involving data collection and labeling, feature scaling, model selection with hyperparameter tuning, model training and validation, model testing and deployment. The model used is a 1-D Convolutional Neural Network (1DCNN) architecture with 1 convolutional layer, 1 pooling layer and 3 fully-connected layers, achieving 97.15% accuracy. To address energy limitations of wearable processing, several quantization techniques are explored and their performance and energy consumption are analyzed. We propose a novel Spiking-Neural-Network(SNN) based respiratory classification solution, which can be implemented on event-driven neuromorphic hardware. We propose an approach to convert the analog operations of our baseline 1DCNN to their spiking equivalent. We perform a design-space exploration using the parameters of the converted SNN to generate inference solutions having different accuracy and energy footprints. We select a solution that achieves 93.33% accuracy with 18 times lower energy compared with baseline 1DCNN model. Additionally the proposed SNN solution achieves similar accuracy but with 4 times less energy.