After working with support on this, I found that the current version of UltraEdit 13.10a+2 does not have support for HTTP proxy support for ftp or ssh connections.
Just to let you know, I just downloaded the current version (v13.20). At least this version does support HTTP Proxies for SSH connections. For whatever reason, it doesn't suppor HTTP Proxies for SFTP connections, which I would have thought would be a simple step after getting SSH to work.