Theoretically, yes, you can. But it is impractical.
Both you and your opponent need to click "Start" within 30 seconds of each other, and you need to know who is hosting and who is the guest. The guest needs to know the IP address of the host to connect to.
Think of it like a sniper rifle and a scope. You don't technically "need" the scope to shoot a small faraway target, but you just won't hit it reliably without one. Hamachi is one good scope, but there are others you can use.
As I said, you need to talk to your prospective opponent in real time, and to know your/their IP address. Redshark does not provide such functionality. Hamachi does. You can replace Hamachi with other, similar services, but you do need the type of service it provides.
Hamachi does not intertwine with Redshark or anything like that. It's just a chat room program that gives you an IP address. You can use any other chat room instead if you like, but you do need a chat room.