AI assistants such as Alexa, Google Assistant, and Siri, are making their way into the healthcare sector, offering a convenient way for users to access different healthcare services. Trust is a vital factor in the uptake of healthcare services, but the factors affecting trust in voice assistants used for healthcare are under-explored and this specialist domain introduces additional requirements. This study explores the effects of different functional, personal, and risk factors on trust in and adoption of healthcare voice AI assistants (HVAs), generating a partial least squares structural model from a survey of 300 voice assistant users. Our results indicate that trust in HVAs can be significantly explained by functional factors (usefulness, content credibility, quality of service relative to a healthcare professional), together with security, and privacy risks and personal stance in technology. We also discuss differences in terms of trust between HVAs and general-purpose voice assistants as well as implications that are unique to HVAs.