Issues associated with a lack of diversity and inclusivity persist in the domain of free/libre and open source software (FLOSS) development and in software development generally. Researchers have suggested that the corporatization of FLOSS affords opportunities for creating an inclusive workforce. To understand the potential for firms to increase diversity, we conducted a mixed-methods study of diversity and corporate engagement in FLOSS projects. We integrate the results of a qualitative survey and a big data analysis to understand developer perceptions of corporate engagement and its association with gender and geographic diversity. In the qualitative component, we collected responses from 64 FLOSS contributors to elicit their perspectives on corporate engagement. In the quantitative component, we analyzed GitHub data from 38 projects and 9,990 contributors to investigate differences in participation and diversity based on corporate engagement. We find that contributors vary across dimensions that can inform diversity interventions: stances toward corporations and orientation towards individual/collective benefits. Our results suggest that corporate engagement may limit a project's contributor base and geographic diversity. Yet, organizations subsidizing FLOSS have opportunities to increase access to projects which would benefit diversity. This research serves to identify individual and organizational factors which may harm and help diversity initiatives.