Semantic communication has emerged as a promising paradigm for enhancing communication efficiency in sixth-generation (6G) networks. However, the broadcast nature of wireless channels makes SemCom systems vulnerable to eavesdropping, which poses a serious threat to data privacy. Therefore, we investigate secure SemCom systems that preserve data privacy in the presence of eavesdroppers. Specifically, we first explore a scenario where eavesdroppers are intelligent and can exploit semantic information to reconstruct the transmitted data based on advanced artificial intelligence (AI) techniques. To counter this, we introduce novel eavesdropping attack strategies that utilize model inversion attacks and generative AI (GenAI) models. These strategies effectively reconstruct transmitted private data processed by the semantic encoder, operating in both glass-box and closed-box settings. Existing defense mechanisms against eavesdropping often cause significant distortions in the data reconstructed by eavesdroppers, potentially arousing their suspicion. To address this, we propose a semantic covert communication approach that leverages an invertible neural network (INN)-based signal steganography module. This module covertly embeds the channel input signal of a private sample into that of a non-sensitive host sample, thereby misleading eavesdroppers. Without access to this module, eavesdroppers can only extract host-related information and remain unaware of the hidden private content. We conduct extensive simulations under various channel conditions in image transmission tasks. Numerical results show that while conventional eavesdropping strategies achieve a success rate of over 80\% in reconstructing private information, the proposed semantic covert communication effectively reduces the eavesdropping success rate to 0.