Electronic – Can a transistor send negative voltages when used as a switch

I have used relays as a switch in order to turn on and off things. This is how I do it:

My IC is my integrated circuit call it arduino. With a small current I am able to turn on and off things and I can now replace mechanical switches such as this one with a transistor.

Anyways now my question is will this also work when sending negative voltages? We have 2 different internet providers meaning we have 2 separate cat6 cables. Each provides internet. If the internet goes down from one cable then we connect the second Ethernet cable. I just learned that cat6 cables send voltages between -2V and +2V. Since the cat6 cable has 8 wires inside I was wondering what will happen if I do the following for each of the 8 cables inside the cat6 cable:

1. Cut the cable at the middle.
2. Connect one side to the collector of a transistor
3. Connect the other side to the emitter of the transistor
4. Connect the base to my arduino giving it a small current so that the switch is closed.

If I where to do this will there still be an internet connection? If there is an internet connection then I can use this mechanism to switch between providers instead of having to change the cables manually? I will probably not do this. I am asking for purposes of learning. I do not want to use a relay as they are larger and using 8 for each cat6 cable will be a lot.

Edit

The transistors will not be switching on and off at the frequency of Ethernet. They will be either on or off. They will only change states when switching between Ethernet cables. If they are not able to transmit negative voltages can I use 2 transistors in reverse direction like so:

That will represent the connection of 1 of the 8 cables of the cat6 cable. The base pin of the transistor is on the center. That will always have a possitive voltage to leave the switch on. Because there are 2 transistors in reverse direction it will now be possible to send negative voltages no?

Edit 2

This will represent one of the 8 cables illustrated with a better diagram:

Using something like this I should be able to send signals from both directions correct? I plan on leaving the transistors on until I want to change internet service providers They will not be turning on and off rapidly.

There is a problem though. I can connect the base to +5v of arduino but how can I share the same ground so that this diagram works?