Hi there,
Sorry to hear about your troubles. Generally if you're getting disconnected like that, it's going to be a matter of the amount of data flooding a network adapter or similar. The first job is to each make sure that you have the very latest version of your network card drivers -- sometimes that can solve it right there.
The next thing to look at is the MTU, or Maximum Transmission Unit, on the game settings window. It is set by default at 1400, which is a very standard sort of value for games, but can be too high for some routers or ISPs. There is a more detailed article about this elsewhere, but basically what that causes is a bunch of packet splits and wrappering when it goes through the network.
Almost always Hamachi would avoid this sort of problem, though, so that's kind of curious that it would be an issue -- I don't think I've ever heard of this issue happening through Hamachi, to be honest.
Anyway, you can turn down the Maximum Transmission Unit on both of your machines to be something like 1200 or 1000. The values do not have to match between your machines, but it doesn't hurt if they're in the same ballpark for purposes of just testing out the network.
Honestly since you are having problems even through Hamachi, I am thinking that there is a network card that is having a buffer overflow or something, so a driver update might be all that is needed.
I hope that helps!
