Recent research is emerging in the field of Social Robotics where robots have the potential to serve as tools to improve human well-being. However, research exploring the expectations and perceptions of prospective users of such robots, and the professionals who currently deliver these interventions, is limited. In this paper, we present qualitative analysis of discussions with prospective users and experienced coaches regarding the design of robot well-being coaches. We invited participants interested in well-being practices to take-part in a Participatory Design (PD) study, consisting of individual interviews and a focus group discussion (NP = 8). Discussions focused on ideating how a robot could function as a mental well-being coach, based on their experiences with well-being practices. Data triangulation was employed by interviewing three professional coaches as additional sources of information. This resulted in a rich set of data, which we transcribed and analysed using Thematic Analysis (TA). The developed themes regarding robot features, form, behaviours, robot-led well-being practices, and the advantages and disadvantages these could provide, were compiled and are discussed in detail. We present this data together with tabulated quotes from the participants and coaches, to pave the way towards designing robot coaches that can provide supportive interventions to improve the mental health and well-being of their users.