If you have government contracts, which is the way SpaceX survives, the rule is you have to consider everyone for employment that can be employed. That means you have to consider refugees as long as they can be legally employed. You don't have to employ them but you can't exclude them due to their status.
Could you elaborate further on what you're saying? I've read the article, seems like standard security clearance stuff for sensitize material. Musk didn't become a citizen of the US till 2002 if what I'm reading is correct and I don't think they have a problem hiring staff from outside of the US.
Functionally you can't do duties regulated by ITAR without authorization from the U.S. Department of State... so effectively you can't fill that position. While looking for jobs I've applied to some US companies who are involved in defense contracting before for a non-defense position and they basically label me (a Canadian) as a foreign person at the first screening and they don't call me back.
For practical purposes, ITAR regulations dictate that information and material pertaining to defense and military-related technologies (items listed on the U.S. Munitions List) may only be shared with US persons unless authorization from the Department of State is received to export the material or information to a foreign person. US persons (including organizations; see legal personality) can face heavy fines if they have, without authorization or the use of an exemption, provided foreign persons with access to ITAR-protected defense articles, services or technical data.
I'm saying it's common practice for companies in the US to "consider" candidates they have no intention of actually considering, because they already know who they want (or what 'type' of person they want) and that these are hoops companies jump through that don't end up doing the things people expected the hoops to do.