The following commands were run to gather diagnostic information about the issue:
netsh interface httpstunnel show interfaces
Last Error Code: 0x2af9
Interface Status: failed to connect to the IPHTTPS server. Waiting to reconnect
This error was caused by a proxy server set on the workstations in question in Internet Explorer. When a proxy is set, Direct Access will attempt to create the IPHTTPS connection through the proxy server. If you want to leave proxy servers intact in Internet Explorer properties, what you can do is add vpn.example.com (the connection URL) as a proxy exception in Internet Options.