I would think Trump could fire them, not sure what, if any, there is in place to protect them.
The thing is, if they honor their oath in the slightest they wouldnt cooperate. Its a huge security breach to have one person have access to the complete list of federal employees, huge huge. It would be the first thing a dictator would have to do, get the list of everyone, then you can target and force people to cooperate. The whole point of all this security and compartmentalization is specifically this very security risk. So any director that actually lets their department answer, and directs their people to answer, would be putting the USA at risk and violating appropriate security protocols if nothing else.