Next-generation networks aim for comprehensive connectivity, interconnecting humans, machines, devices, and systems seamlessly. This interconnectivity raises concerns about privacy and security, given the potential network-wide impact of a single compromise. To address this challenge, the Zero Trust (ZT) paradigm emerges as a key method for safeguarding network integrity and data confidentiality. This work introduces EPS-CNN, a novel deep-learning-based wireless device identification framework designed to serve as the device authentication layer within the ZT architecture, with a focus on resource-constrained IoT devices. At the core of EPS-CNN, a Convolutional Neural Network (CNN) is utilized to generate the device identity from a unique RF signal representation, known as the Double-Sided Envelope Power Spectrum (EPS), which effectively captures the device-specific hardware characteristics while ignoring device-unrelated information. Experimental evaluations show that the proposed framework achieves over 99%, 93%, and 95% testing accuracy when tested in same-domain (day, location, and channel), cross-day, and cross-location scenarios, respectively. Our findings demonstrate the superiority of the proposed framework in enhancing the accuracy, robustness, and adaptability of deep learning-based methods, thus offering a pioneering solution for enabling ZT IoT device identification.