Hmmmm, I am testing something to see if it fixes my problem, but I had a thought about a possible source of the problem.

Looking at the properties for my network adapter, I see the QoS packet Scheduler. I didn't think about it sooner, but since this in theory prioritizes packets, it could be that it is the source of our problems. So, I did what anyone would do to test, remove the check from the box, and saw everything drop since the scheduler was being used, so it would reset all connections.

I will report if this stops the problem, but since the problem happens at random, it may take a few days/weeks to be sure if the problem is solved. My gut instinct here(as an old-school technical person) is that the QoS could screw things up for mIRC as well as other programs(giving priority to things that no one really cares about).