When using LLMs to address Non-Functional Requirements (NFRs), developers may behave differently (e.g., expressing the same NFR in different words). Robust LLMs should output consistent results across these variations; however, this aspect remains underexplored. We propose RobuNFR for evaluating the robustness of LLMs in NFR-aware code generation across four NFR dimensions: design, readability, reliability, and performance, using three methodologies: prompt variation, regression testing, and diverse workflows. Our experiments show that RobuNFR reveals robustness issues in the tested LLMs when considering NFRs in code generation. Specifically, under prompt variation, including NFRs leads to a decrease in Pass@1 by up to 39 percent and an increase in the standard deviation from 0.48 to 2.48 compared to the baseline without NFRs (i.e., Function-Only). While incorporating NFRs generally improves overall NFR metrics, it also results in higher prompt sensitivity. In regression settings, some LLMs exhibit differences across versions, with improvements in one aspect (e.g., reduced code smells) often accompanied by regressions in another (e.g., decreased correctness), revealing inconsistencies that challenge their robustness. When varying workflows, the tested LLMs show significantly different NFR-aware code generation capabilities between two workflows: (1) integrating NFRs and functional requirements into the initial prompt and (2) enhancing Function-Only-generated code with the same NFR.