As electronic HID ballasts operates the lamps at square wave current and at a higher frequency than the mains, I'm wondering if an EOL cycling lamp won't be able to extinguish, because it can stay live longer because of the operation wave and the frequency of the ballast?

They can but good ballasts recognize the increase in arc voltage, and cut out.

Because HPS tends to rectify before extinguishing, as far as I've seen in one of my meeting with Ash.

Some do, some dont, and you dont want the ballast to randomly cut out on a lamp that is a little rectifying but still works

It probably cannot be detected reliably (and without false positives) before the lamp goes out on its own at least once

For the rectification cutout it does matter what is the exact ballast output topology, whether it can deliver DC current or not. If yes (e.g. when using full bridge output stage), there is no need for rectification cutout, rectification is not endangering anything. And given the square wave frequency (150..400Hz), even when it lights only for one halfwave, the resulting flicker won't be that a problem.
But if the output stage is a halfbridge against a center point between capacitors, the ballast must shut off when the center point is imbalanced too much in order to protect those capacitors from overvoltage (they are designed so the bus voltage is split among them, imbalance causes one of them getting way higher voltage and so may exceed its rating).
Some halfbridge ballasts may address this by just adjusting the duty ratio of the square wave, so if one polarity yields lower current, it is let to flow the longer time and maintaining the center voltage this way. Then there wil be some limits set how far the duty ratio may go, shutting down when that would be too much.

In the real life, all electronic ballasts sharing this topology, half bridge + split capacitor, that I have on hand, Philps's and Osrams, have their capacitors significantly up-rated by voltage. 315-350V caps for expected Vbus of around 400V. So even a serious imbalance is *not a problem*.

And of course, all modern electronic ballasts are driven by some microcontroller brains, so in the case the lamp attempts to cycle, they will shut down after a programmed number of tries, also the most sophisticated ones will report this by DALI, for example.
 

HID lamps can invariably rectify at times and this is normal - especially at starting. Of course the ballast must be able to withstand that, without damage and without going into protection mode. Perhaps that is taken into account with the higher rated capacitors to an extent

In PC power supplies using the old half bridge circuit (where the outcome of any imbalance would be saturation of the transformer) they solved this with a film capacitor in series with the transformer. Is this present in any HID ballasts ?