Do airplanes have a maximum headwind gust component when landing?
This is quite similar to the linked question but does have some differences. Gust components while reported differently are still part of the crosswind component. In other words a reporting like 


  Rwy 03, wind 090/25 gusting 40 kts


Is another way of saying the wind is holding at 25Kts and could be as high as 40Kts at times while you are on approach. In other words you may have a 25kt cross wind on touch down or you may have a 40Kt cross wind. Generally speaking gusts occur in the direction of the wind. In some cases gusty days also come with shifty wind conditions which add for a whole new element of fun. 

The maximum gust component is not quoted in the POH because there are to many inconsistent factors that may effect it. The main one being the duration of the gust. A short gust well in excess of your crosswind maximum may be recoverable while long gusts in excess of your crosswind maximum may be unsafe. Strictly speaking a loose limit would look something like 

(maximum reported crosswind) + (gust wind * gust time) > cross wind component 

but gust time is not reported and thats a lot of trig to crunch on the fly so no limit is typically given. I have landed a Piper Archer in conditions gusting in excess of 40Kts safely but the wind was no more than 20 degrees off the runway. 

Eventually you just run out of control surface... 


  Example: imagine a GA aircraft with a typical takeoff speed of 60 kts,
  approach speed of 70 kts. Runway is 27, wind is 270 degrees at 20 kts.
  What would be the maximum gust speed along the runway to operate in
  safe conditions?


In this scenario, with the wind down the runway, your maximum gust component is 50Kts. Above that and at your 70kts approach speed you would be moving backwards across the ground making it quite hard to reach the runway and potentially difficult to land the airplane if a gust were to occur at a low altitude. Most trainer aircraft like the one you describe cant do better than ~120Kts so 100Kts is about your gust limit otherwise its going to be hard to get to the runway.  

NOTE: Gusty conditions (in my experience) are also generally accompanied by turbulence and in bad cases, wind shear. In many instances all of these factors combined can make an approach unsafe or un-flyable. 

There is also an interesting discussion on it here.