In General, No
Not all federal executive employees can be fired by the President. Historically, they used to be able to. However, this led to a spoils system in which new Presidents would fire large portions of the executive branch and replace them with their own supporters.
In 1833 the Civil Service Act was passed to limit this kind of behavior (the act was revised extensively in the 1970's). This report to Congress describes limitations on firing civil servants (see pg. 27). Employees may be fired for performance or behavior-related reasons, but not for non-employment related reasons. There is an appeals process to a third-party for disputes over personnel actions.
The Special Position of the Dept. of Justice
The Department of Justice is also in a special position. There is a lot of media coverage on the subject. Time published this article summarizing the President's relationship to Department of Justice officials here, which is based on Dr. Harriger (prof. of political science at Wake Forest). The high-level synopsis is that since Departmen of Justice investigators have a Constitutional duty to ensure that the law is executed correctly, they cannot be fired by the President for doing that job.
Of course, what is allowed in practice will be a highly political (as well as legal) situation.