This work introduces a novel methodology for the development of interpretable reduced-order dynamic models specifically tailored for jumping quadruped robots. Leveraging Symbolic Regression combined with autoencoder neural networks, the framework autonomously derives symbolic equations from data and fundamental physics principles capturing the complex dynamics of jumping actions with high fidelity. This approach significantly reduces model complexity while enhancing interpretability, facilitating deeper insights for legged robotic applications. The efficacy and accuracy of the proposed models are validated through comprehensive experimental studies, marking a substantial advancement in the design of agile and efficient legged robots. This research demonstrates the outperformance of a learned 2D model compared to existing template models such as the ASLIP. Also, an analysis of the dimensionality of the learned model is conducted showing the overarching tradeoff between accuracy and complexity. The method is validated on different simulated quadrupeds and an actual hardware robot.