IMO crystals are ok. We find new materials every day, so crystals with exotic properties are perfectly plausible. I know I'm digging too far and frankly further than my actual knowledge of physics, but... those particles that would be gathered with crystals seem less plausible to me. Not their existence but their ability to propel matter FTL. For instance - tachyons are theoretically possible particles that would, should they exist, always move FTL. The less energy they have, the faster they move.
So, let's say they exist and we have crystals to collect them. Somehow they are trapped in that crystalline structure and so forth. That does not help with moving any other matter FTL. You would need to find a way to transform the regular matter into tachyons to move FTL.
Thinking about it, there might be an explanation... These particles could be naturally "producing warp bubbles" and you need a lot of them to fit a ship into their bubble. Hmmm... like photons can be massless despite the Higgs field. No other particle known so far is without interaction with Higgs field, so one could say photons have their own anti-Higgs bubble. These particles might be moving through space in a warp bubble and are not interacting with the space at all.
Then again, one can't bathe in photons and be massless. Probably.
My head is starting to hurt.
Bah, gone waaaaaay over my head. Please, physicists, tear this apart and find a better solution.