FRAM is highly radiation resistant. Therefore you won't be able to erase them with X-rays like flash memory.
(ionising radiation doesn't affect magnets)
FRAM is ferroelectric, not ferromagnetic.
http://en.wikipedia.org/wiki/Ferroelectricity
http://en.wikipedia.org/wiki/Ferromagnetism
Not sure why I started going on about magnets, obviously I wasn't thinking straight. But the point still stands, the FRAM storage is not susceptible to radiation (as charge polarisation is not affected), only the control circuitry is.
X-rays are comparatively low in energy, generally below 1 MeV. The really energetic particles that can disrupt memories come from radioactive decay or cosmic rays. So-called "rad hardened" chips used for spacecraft need to withstand "single event upsets" (SEUs) from multi-MeV particles. ECC memory helps a lot.
ECC does help quite a bit, but it does have its limits. From what I've seen, rad-hardened electronics tend to come in two flavours, those with larger features (so the ionisation is a lower percentage) and redundancy (i.e. ECC, triple redundancy and voting logic).
addendum: I searched for more info on this topic and it looks like very low energy X-rays are actually the ones most responsible for leaking charge off the floating gate, in what I presume to be a similar process to UV erasing an EPROM. At these low energies disruption to active circuits (sense amps etc) is unlikely.
(http://www.spansion.com/support/application%20notes/x-ray_inspection_on_flash_an.pdf)
I know that the south Atlantic anomaly causes a lot of hassle for satellites (with cubesats, containing non-rad-hardened components, being reset or having corrupted memories when passing through it). But the Van Allen belts are quite high energy (1 - 100 MeV). But as you say, low energy radiation (i.e. UV and x-rays - remembering that UV and soft-x-rays cross over) can cause a lot of problems, but it is very dependant on the environment you are in and what you are using (I think analogue electronics will just have higher noise).
The way that UV EPROM is erased is exactly the same effect (have a look at the photo-electric effect on wikipedia). UV will erase flash memory as well, but you have to de-cap it first. Some microcontrollers have metal guards over there write-once fuses to stop someone from de-capping them and then reading the data off.
On a bit of a side note, a lot of servers are now starting to see effects from background radiation. With the increase in memory density, and the decrease in feature size, the amount of charge on each floating gate in a RAM module is smaller, requiring less radiation to flip each bit, making them more susceptible.
On an even more of a side note, but still kind of related (and will hopefully make up for my magnet screw-up earlier), the feature size of flash memory could be reduced further than it already is, but with the reduction in size comes an increased tunnelling effect, which reduces the data life of the memory. The thousands of years normally quoted is in part due to background radiation exposure, but a large proportion of it is the probability of the charge tunnelling off the floating gate. It is in part why memory manufactures are looking at ways to stack memory on top of each other inside a chip (which has just reminded me of Hitachi's get perpendicular video).