No, that won’t happen. That is, I wouldn’t put it past the GOP to pull a stunt of asking or demanding she sign one, but she won’t because she doesn’t need to because she never had to in the first place.
The secretary of state for the US is NOT “an employee”. That role is a CABINET position. The holder of that cabinet position is bound by the oath of office. A person who has been nominated to serve as a cabinet minister and had that appointment approved by the Senate then serves at the pleasure of the president and otherwise subject to IMPEACHMENT proceedings in Congress.
Meanwhile, no “employee” of any federal government agency is subject to dismissal at the pleasure of the president or is otherwise subject to impeachment proceedings by Congress.
This is not rocket surgery, folks. This is Fox thinking you all are dumb as rocks.