Raspberry Pi controlled power on/off switching of an Ethernet connection
Sorry but your circuit won't work.  As Michael says Ethernet is too sensitive for solutions like that.  So here are a few other suggestions that might help.

The simplest solution would be a managed switch that both machines were connected by and then use the Pi to connect to the management interface, via serial or telnet, and tell the switch to close and open the ports as you need.

Or write and install a service on the hosts that when the authenticated Pi sent a message to them the services used operating system functions to disable the network interfaces.  Obviously there is a problem with this solution when it comes to turning the connection back on.

You might be able to use a USB Ethernet adapter on the Pi to create your own two port switch, then bridge the connections using the Pi's OS.  The Pi can then enable and disable the connections at will, but I'm not sure what the performance would be like.