Nothing to do with driving style and everything to do with the individual gearboxes themself. How they react to a problem depends entirely on how the fault manifests itself. For example McLaren said they'd had the same problem before with gearboxes in the past with the same warning but had been able to get through the race without it breaking down, which is why they were surprised at the DNF, so it makes sense that JB could make it through the race unscathed but still need a gearbox change. Similarly last year both Lewis and Vettel both had gearbox failures but Lewis' resulted in him losing gears cumulatively before having to retire while Vettel was able to continue albeit at a slower speed.
Vettel does not drive a McLaren. In your example Button did not have any gearbox problems.. and there is nothing to suggest that they both had the same problem. (Although the rumor at the time was that Vettels gearbox had Webber-winitis)
In this particular case it appears that the defect was introduced in Singapore, as it appears to have affected both cars.
If this "problem" was not seen during or post Monza inspection then the gearbox itself would only be a normal gearbox going into only into its second cycle.
So as far as I can see they both started with normal intact gearboxes. The chances are the damage was caused solely by this specific problem at Singapore.
The idiosyncrasies of their driving may or may not have influence on the mechanical "well-being" of the cars.I believe they do. However physical nature does care about belief, so in the end our beliefs are irrelevant.
Certainly the root cause of this particular problem was not in the either drivers hands and we don't know defect was inherently more "pronounced" on one car than the other.
we also don't have enough information to know whether Lewis could have lasted longer in the race if he had not made the mistakes in qualifying, or perhaps less rough riding over the curbs but in the end he suffered a huge points loss.
Given that all things should be equal, the chances of a driver having gearbox problems five times and his teammate not are 1/32. Perhaps its just down to luck, or it could be an incompetent mechanic? or is it something the driver is doing?.
At Singapore one car got to the end, the other only half way.
Luck? .. perhaps, perhaps not.http://www.f1complet.../view/1286/389/