Any company that has employed someone to carry out a task or tasks has basically made an agreement that the employed will be paid to complete the required task or tasks. The employee then has agreed to carry out the task or tasks in order to receive payment. This agreement can be verbal or written, but is legally binding.
It is then within the employers power to specify exactly how the work is to be done and require the employee to follow the instructions given, or it can be considered a breach of the initial agreement and the employment can be terminated at any time, providing that the person employed has not been asked to do something illegal, or unsafe by regulated standards.
Yes, the employer does have the right to dictate how a driver uses their logbook, as long as it does not interfere with the regulated standards required by FMCSA, or put the driver in harms way.
The driver also has the legal right to terminate employment at any time for any reason, though there may be civil penalties if it causes great loss to the employer due to a violation of the work agreement.