Can tailwind produce reverse-thrust?
Only when on the ground, and at the inlet end, where a strong wind from  the side and behind could influence the engine's ability to draw air in the front at low power settings.  Some engines have limitations or restrictions on how power is applied during takeoff in strong tail or quartering winds, because of initial flow disruptions at the front from air having to make a sharp corner into the intake, where large thrust increases could cause compressor stall/surge.  

You might have a procedural requirement to advance power to some intermediate setting, then let it stabilize there before going the rest of the way to takeoff thrust. The GE CF-34 has limitations like this, where winds above a certain speed and off angle a certain amount require a staged application of thrust.

Once you're in the air, there is no such thing as a tailwind as far as the airplane is concerned except insofar as tailwinds add to ground speed.  For the engine, it's all coming from the front as soon as you are moving with any significant airspeed.