Embodied intelligence is shifting artificial intelligence from passive digital perception toward active physical interaction. However, foundation-model-enabled embodied agents face a fundamental tension between open-world cognition and resource-constrained deployment. On-device models are limited by computation, memory, and energy budgets, whereas cloud-centric solutions introduce latency and reliability risks over dynamic wireless links. Edge general intelligence provides a promising cognitive backbone, but existing frameworks still lack physical grounding, action awareness, and mechanisms for actively acquiring useful physical experience. To address these limitations, this article introduces edge-native embodied intelligence (ENEI), an action-aware wireless edge framework that integrates embodied agents, the 6G communication and networking fabric, and edge cognitive services into a 6G-mediated bidirectional edge-embodiment loop. Along the edge-to-embodiment axis, confidence-aware assistance and edge-driven generative adaptation enhance local autonomy under out-of-distribution (OOD) conditions. Along the embodiment-to-edge axis, value-of-experience guided active embodied federated learning enables physical actions to generate informative experience for continuous edge model evolution. The 6G fabric supports both directions through goal-oriented transmission and programmable radio-resource allocation. Two case studies on OOD drone navigation and mobility-driven federated learning illustrate the feasibility and communication efficiency of the proposed mechanisms. ENEI provides a unified perspective in which edge cognition strengthens embodied action, while embodied agency actively enriches edge cognition, laying the foundation for scalable, adaptive, and self-evolving embodied wireless systems.