I’m shocked that I actually found the answer to this on the EpicCare Knowledge Base. Should have looked there as well before posting!
Edit the web.config of the site in question. In the <serviceCredentials> element, uncomment the serviceCertificate element and/or enter the certificate name in the findValue attribute.
<serviceCredentials>
<!–
<serviceCertificate x509FindType=“FindBySubjectName” findValue=“localhost” storeLocation=“LocalMachine” storeName=“My” />
–>
</serviceCredentials>
Remove the <!-- that is above and the --> that is below the serviceCertificate line. Save and try connecting again.