Could a plane land vertically in a strong headwind?
Yes.

I assume that you do not have a good understanding of aerodynamics.  I apologise if that is not the case.

The way to think about these sort of questions is to ask "does the aircraft know that it's moving in relation to the Earth".  The answer to this is "no".

The only thing that matters, and the only thing that the aircraft cares about is airflow (wind) over the wings.

Ignore all sorts of little side effects and imagine an aircraft needs 70 kph of wind over the wings to generate lift and stay afloat, straight and level.

Now, imagine the aircraft is flying along and you could somehow make the Earth disappear but leave behind gravity and the atmosphere.  What would happen to the plane?  Nothing!  It still has 70 kph of wind over the wings and it keeps flying.

Now put the Earth back.  It still keeps flying.

If the pilot reduces the power to 69 kph ("thrust X"), the force generated by lift pushing the aircraft up is a little less than that generated by gravity pulling the aircraft down and the craft will start to descend.  Eventually, it will land (assuming that you remembered to put the Earth back!).

So two scenarios.  First, one where the air is perfectly still and the thrust from the engine ("thrust Y") is pushing the aircraft forward at 70 kph.  Second, where the wind is blowing at 70 kph and thrust Y is such that the aircraft is stationary with respect to the Earth.  Again, the aircraft does not know, or care, that it is stationary with respect to the Earth, just that it has 70 kph of wind over the wings.

Modify the second scenario slightly so that the wind is now 69 kph. The aircraft moves forward at 1 kph. Again, the aircraft does not know, or care, that it is going fowards with respect to the Earth. The wind over the wings is still 70 kph (speed of wind, 69, plus speed of aircraft 1). 

Reduce the thrust to X again and the aircraft will start to descend, and also become stationary.  Eventually, it will land with no forward or backward speed.

In reality, this would be almost impossible since the wind is never of a constant speed from a constant direction so little movements would always be taking place.  

I have seen a video, which I can no longer find, of a bi-plane which took off into a strong wind, flew backwards down the runway and landed on a spot before the one it took off from.

This comes very close.  With just a little more wind, a vertical landing would have been possible.