What type of switch should I use?
A MosFET doesn't have a volt drop across the gate, but a resistance, so that would be my first thought. You obviously need an active circuit to control the switching FETs. A comparator would be my first choice to detect the Red voltage and boost it to a level sufficient for logic operations and thus control the Mosfet gates.

When Red is detected as high, switch the Red FET on and Green FET off, when it's low, Red off and Green on. You would want to select a mosfet with a low Drain-Source resistance.

This is assuming that the lines are fully on or fully off, that is only switching high and low as implied by your question.