I think it MIGHT be possible, but definitely a lot more to it than your typical A/B switch. The main reason is that the RJ45 connector has 8 separate wires. I am not aware of an 8PDT stomp-style switch and have VERY serious doubts that such a switch exists. You would have to use a DPDT connected to a relay. This gets a bit complicated, as you would need to power that relay somehow. I suspect the whole thing would be more trouble and expense than it'd be worth.
Alternately, you could just plug one Variax in through the VDI input and the other through the 1/4" guitar input. Then, set up duplicate patches on the HD500X, with one set using the VDI and the other set using the 1/4".