So far, only the White House has been claiming Iran. I wouldn't put much confidence in that unnamed "US official".
As for the technical plausibility, perhaps. I don't know how much storage they need to operate those things. If they took the cheap way out and put Linux on an SBC and patched in sensors and controls, it's quite possible; if it's a custom system (the more likely case, most militaries want something a bit more robust), who knows. Flash is certainly used, but they might keep the particulars of targeting and flight planning in SRAM (or less likely, DRAM?), and this would be one motivation behind that. Flash might still be used to hold terrain data, say; and if there's a limited amount available (so-and-so many megs, not gigs of storage), they might only load the local area, or at least at high resolution. That could still be whole quadrangles of the world though, not real helpful at nailing down any particular base.
Looks like the wreckage is fairly intact, so even just desoldering the chips may prove fruitful. Flash can be decapped and probed intact, if worse comes to worst. Fractured chips aren't so useful though.
Tim