Don't worry about it
The Pathfinder Unchained summoner, like the original summoner, has the following text describing the class feature eidolon:
The eidolon forms a link with the summoner, who forever after summons an aspect of the same creature.
Unless an effect's severed the link the eidolon forms with the summoner, the summoner brings forth only an aspect of the creature to whom the summoner's linked, not the actual creature to whom he's linked, just like a normal summoning spell (albeit with vastly increased complexity), not a calling spell. This makes permanent removal of the eidolon class feature a plot issue not a rules issue. (That is, the creature to whom the summoner is linked is never given a precise location or statistics, so what's preventing the eidolon from sending eidolon aspects when the summoner asks is GM fiat or specific effects that prevent summoning.)1
In other words, the linked eidolon can't die unless the DM just says such a thing happens. And unless that happens, the eidolon aspect probably can't permanently die either, but there's some debate in Pathfinder Society if an ability damaged or diseased eidolon that's resummoned remains ability damaged or diseased.2
I searched Paizo's Pathfinder Society messageboards for further information about this topic and found nothing addressing it directly (and it appears even eidolon deaths in combat are uncommon, so eidolon deaths by GM fiat even moreso?). However, if planning for long-term play, this thread and this thread wondering about tracking eidolon's injuries from scenario to scenario might be of interest.
1 A longer and even more extreme version of eidolon aspect destruction ("Your eidolon is stuck 100 years in the future," said DM Nelson. "Ha ha!") that I believe can still be remedied is detailed here.
2 I would argue a lot that The eidolon doesn't heal naturally applies only to hp, as that's the statement's context. Applying that to everything seems deeply against the statement spirit, and literally means, for example, any psychological trauma the eidolon suffers never heals, and that's just sad.