I don't have the faintest clue on how a software or hardware load balancer works. I guess the hardware load balancer is basically a switch and based on some algorithm decides which node to switch to for a incoming request. On the software load balancer front, I guess the software picks up a node and uses a reverse proxy connection to it. In such a scenario, 2-way SSL wont work as the load balancer cannot have the client's private key.
Again, I don't how a software load balancer works but as my application would need a load balancer and as the application uses 2-way SSL connection, I wanted to know how does a software load balancer take care of a 2-way SSL connection.