Computer hardware engineers require atleast a bachelor's degree in computer engineering. A degree in computer science and electrical engineering are also accepted. Although computer hardware angineers work more on hardware components, they must also have a familiarity in computer programming. You can obtain this by taking computer science courses.
Some employers prefer and/or might require accreditation from the Accreditation Board for Engineering and Technology (ABET). Graduating with accreditation gives you more chance to get accepted for a job.
Also, some firms may require you to have a master's degree in computer engineering for a higher position and salary. Some engineers also obtains master's degree in business because they might someday take over the firm and become the CEO. With the fast paced development of technology today, an engineer must keep up and continue learning about technologies.