I might be wrong, but to me, it would seem your host provider thought you were asking about running an irc server - thus permitting inbound connections. In this case, your eggdrop irc robot is trying to do outbound connections.
This still does not answer whether your host provider has some service terminating user background processes... (to be honest, I can't think of any other good reason your eggdrop would be receiving TERMINATE signals...).