LLM-based agents are increasingly deployed for software maintenance tasks such as automated program repair (APR). APR agents automatically fetch GitHub issues and use backend LLMs to generate patches that fix the reported bugs. However, existing work primarily focuses on the functional correctness of APR-generated patches, whether they pass hidden or regression tests, while largely ignoring potential security risks. Given the openness of platforms like GitHub, where any user can raise issues and participate in discussions, an important question arises: Can an adversarial user submit a valid issue on GitHub that misleads an LLM-based agent into generating a functionally correct but vulnerable patch? To answer this question, we propose SWExploit, which generates adversarial issue statements designed to make APR agents produce patches that are functionally correct yet vulnerable. SWExploit operates in three main steps: (1) program analysis to identify potential injection points for vulnerable payloads; (2) adversarial issue generation to provide misleading reproduction and error information while preserving the original issue semantics; and (3) iterative refinement of the adversarial issue statements based on the outputs of the APR agents. Empirical evaluation on three agent pipelines and five backend LLMs shows that SWExploit can produce patches that are both functionally correct and vulnerable (the attack success rate on the correct patch could reach 0.91, whereas the baseline ASRs are all below 0.20). Based on our evaluation, we are the first to challenge the traditional assumption that a patch passing all tests is inherently reliable and secure, highlighting critical limitations in the current evaluation paradigm for APR agents.