RTC web works and RTC eclipse doesn't works with restricted public proxy.

The "Proxy1" checks for what type of browser is being used
e.g If Internet explorer older then 11 is being used then the proxy blocks it.This is usually done by checking for the "User-Agent" field in the HTTP header.                                                                
When you use RTC Eclipse Client the "User-Agent" is set to "Proxy2" Commons-HttpClient/3.1". So this user agent should be added to the " Proxy1" to allow the RTC eclipse client to connect.