Napoleon III's politics in Germany were about two things : first, taking scraps whevenever possible as long as it didn't antagonized too many people in the same time, and second to prevent the rise of a german hegemony especially along the Rhine.
It was why French état-major actually discussed, in the eventuality of an Austrian victory,
if supporting Prussia wouldn't be the best choice. That said, I don't see this support being a military one, except if the
politique des pourboires* pays up this time. But I'm not sure if Prussia can afford, politically-wise, a french support on the price of loosing territories along the Rhine : maybe if it's done at the expense of Bavaria, or possibly a support for redrawning borders in Belgium and Luxembourg which will probably antagonize Britain a lot and certainly weights a lot on a Prussian victory, so...that's not really that likely to happen.
If Bismarck agrees, a reasonable offer may be a rough return to
1814 borders, with several adjustments, especially in Bavarian held territories and aqcuisition of Luxembourg (not directly, but with Bismarck not opposing the territory being sold to Nappie III was it was planned in 1867 IOTL)
French intervention is not only not bound to happen (unless gains are certains, which may not be the case) but if it happens, I doubt it would be to support Austria, mostly because Napoléon would both want to avoid an Austrian hegemony in Germany, and could gain more negociating with Prussia for some territories.
*Basically what Napoleon III did in Italy (military support in exchange of territories) and attempted to do in Germany in the late 1860's