Proxy with username and password is not officially supported. It is possible to try the following things, but it is not confirmed, that it will work.
All steps are performed in file C:\Program Files\Quest Software\Toad Edge\ToadEdge.ini.
1) Add the following parameters at the end of the file and try to start Toad Edge again.
-Dcom.sun.net.ssl.checkRevocation=false
-Djavax.net.debug=true
-Dhttp.proxyUser=someUserName
-Dhttp.proxyPassword=somePassword
2) If the 1) does not work, please add the following parameters to the previous parameters.
-Dhttp.proxyHost=<proxyAddress>
-Dhttp.proxyPort=<proxyPort>
3) Repeat the step 1) and 2) with
-Djava.net.socks.username=someUserName
-Djava.net.socks.password="somePassw0rd"
instead of
-Dhttp.proxyUser=someUserName
-Dhttp.proxyPassword=somePassword
STATUS:
Enhancement request TE-14795 has been submitted to Development for consideration in a future release of Toad Edge, to make a more user friendly way of configuring this.