Nowhere near enough details to give a definitive response, but I would have very serious doubts whether two (or four, as needed) of those little things would adequately conduct even 10BASE-T, much less any higher speed, and certainly not POE. In particular there are issues of current capacity, and perhaps even more important are impedance and frequency response to maintain a network connection.
Since we know nothing about the motivation here, we must ask why is such an unusual method required when there are more conventional methods of logically controlling an ethernet connection.
Without any more details than you have offered, I would have to say it is extremely doubtful that will work reliably.