Interactions with voice-based conversational agents (VCAs) in non-humanoid forms are becoming increasingly pervasive, and researches on non-humanoid VCAs engaging diverse human traits have been conducted. However, there have never been studies employing living body's physiological states to be expressed solely through the voice of such VCAs yet. As physiological expressions of such VCAs can have potential for manifesting health-related issues in a human-like way, we selected healthcare scenarios as a case for exploring novel user experiences that they can induce. We conducted design workshops for identifying design considerations and design opportunities for the physiologically expressible VCAs in the healthcare service domain. Following these findings, we designed the new concept of physiologically expressible healthcare VCAs and conducted a Wizard-of-Oz user study. Finally, we summarize the unique user experiences on physiologically expressible VCA's healthcare services and user perceptions of its physiological expressions, and discuss design implications for physiologically expressible VCAs.