It probably is a router or ISP configuration problem, but without you doing a lot of testing to try and find the cause, it would not be easy for us to guess what is causing the problem. Normally, you shouldn't have to make any router changes to connect to IRC. However, if you've made changes, one of them could be causing you problems. You could reset the router to default settings (that will lose all of your forwarded ports and other changes, so only do that if you're okay with having to reconfigure those items). If you reset it, then try connecting before making any changes to the router after the reset.
As for your ISP, you'd have to try contacting them to see if they are blocking anything related to IRC. That said, ISPs rarely tell you if they're blocking stuff and you usually won't even end up talking to anyone who even knows what IRC is unless you can talk them into escalating your call to a higher level tech.
One other option is to enable /debug in mIRC...
/debug @debug
Then, try connecting. Go ahead and do your command to allow the connection to finish. Then, check your @debug window and see what is shown there. If you want, you can paste that here and maybe it will let someone see what's happening.
I'd also recommend unloading all scripts (even if they just message nickserv). That way you know for certain it's not some bug in a script. That's a very easy thing to check and you can always reload them later. It's usually better to test something that's quick and easy right away rather than find out a lot later that the quick/easy thing was the reason and that you spent hours doing other things when you didn't have to.
