Virtual humans (i.e., embodied conversational agents) have the potential to support college students' mental health, particularly in Science, Technology, Engineering, and Mathematics (STEM) fields where students are at a heightened risk of mental disorders such as anxiety and depression. A comprehensive understanding of students, considering their cultural characteristics, experiences, and expectations, is crucial for creating timely and effective virtual human interventions. To this end, we conducted a user study with 481 computer science students from a major university in North America, exploring how they co-designed virtual humans to support mental health conversations for students similar to them. Our findings suggest that computer science students who engage in co-design processes of virtual humans tend to create agents that closely resemble them demographically--agent-designer demographic similarity. Key factors influencing virtual human design included age, gender, ethnicity, and the matching between appearance and voice. We also observed that the demographic characteristics of virtual human designers, especially ethnicity and gender, tend to be associated with those of the virtual humans they designed. Finally, we provide insights concerning the impact of user-designer demographic similarity in virtual humans' effectiveness in promoting mental health conversations when designers' characteristics are shared explicitly or implicitly. Understanding how virtual humans' characteristics serve users' experiences in mental wellness conversations and the similarity-attraction effects between agents, users, and designers may help tailor virtual humans' design to enhance their acceptance and increase their counseling effectiveness.