I've aimed for a clean, professional look they were after, with a modern twist on the tooth-and-cross concept. The blue shades give it a trustworthy feel, while the white cross pops nicely to emphasize the emergency aspect. I've included both a horizontal layout with the full name and a compact version for smaller spaces or app icons. The rounded edges of the tooth make it feel friendly and approachable, which I think will resonate well with patients in stressful situations.
I'm open to tweaks if you've got any ideas to push it even further!