On a sidenote, why is that the case? Sure, SpaceX is a US-based company, but why does that mean that they can't hire non-US citizens with visas etc? What export regulations are they - and come to think of it, what are they exporting?
Many space technologies are listed as munitions in the US, and revealing information about those technologies to foreigners counts as export. There are exceptions, and you can get export licenses (companies routinely do), but hiring non-Americans would be a pain—American employees wouldn't be able to share much information with them!
p.s. Under ITAR, all you need is a green card. I don't know if that's good enough for SpaceX—they might be under other restrictions than ITAR, too.