Healthcare artificial intelligence (HAI) robots could alleviate the current healthcare provision crisis. However, their successful deployment rests on achieving and responsibly calibrating appropriate patient trust in them. Over two user studies, we systematically investigate trust formation in and usage intentions for HAI robots. We first investigate manipulations of visual appearance in a passive observation study without healthcare-specific situational context (n=87) and then in a realistic VR scenario depicting a medical consultation with a HAI robot (n=177) that introduces such situational context (diagnosis and treatment severity). Results show that within a healthcare context, a unique combination of (i) situational (diagnosis and treatment severity), (ii) robot (perceived competence) and (iii) user characteristics (personality and attitudes) determine trust and ultimately influence usage intentions for HAI robots. Furthermore, our results emphasize the absence of HAI visual appearance effects on trust. Our findings support and inform a human centred design approach of HAI robots.