An engineering degree is not required to become a STE. It may help you, but most of the STE's where I worked were former Navy petty officers. They had the advantage of actually being able to operate the plant. The STE is basically the shipyard representative for work that is being done. You'll approve red tags (along with the EOOW) to ensure that systems are ready to safely be worked on and tested. This will require a good knowledge of the plant systems and how they operate, along with the test equipment and temporary systems that may be set up to support the ship, such as power, air, and water (the boat may be in drydock for example). Where I worked, one started out as a Mechanical or Electrical Test Engineer, MTE or ETE. After gaining some experience, one was selected to go to STE school which took about a year and requires passing an exam. In the class you'll learn nuclear theory, systems, etc.
Work wise, it was feast or famine. The rush always seemed to be at the beginning of an overhaul when temporary systems were being hooked up, and then at the end when trying to get the job done. I was in STE school when I left the shipyard for a nuclear power plant. I'm now an SRO which is somewhat similar. We also had an NRC resident inspector who was a former STE. So, there is room to move around, particularly since the skill set is somewhat transferable.