I'll make the argument: No, instructors should not use this tool at Chegg.
In general, Chegg is among the worst actors of the online academic community. Their business model is fundamentally predicated on supporting violations of academic integrity, fraud, and malfeasance. In this sense, we should not be supporting them with engagement, mind-share, or test cases for their systems.
In the same vein of general lack-of-trust, Chegg is not an entity that can be trusted with our exam information. Despite the fact that they currently claim no IP rights to uploaded exams, and a promise to delete information after a month, their promises are not reliable, and we should not give them an opportunity to use our exam information in any scurrilous fashion in the future.
There is no transparency to the exact details of how the tool works. Does it look only for exact text matches, or something similar? If we assume so, then it's quite likely that students will learn to slightly modify the text of submitted questions, so that they avoid the blocking tool, but still get the essence of the answer they need. In this sense, the tool will be a waste of time and give a false sense of security to the instructor.
Additionally, the details say that attempts to post such exam questions to Chegg are simply blocked, with no record of such attempts made. (Per website, "We will not be able to provide any information regarding users who attempt to post a blocked Exam question during the Exam Timeframe.") This short-circuits the possibility of detecting students who make such attempts, and giving required corrections to the students in question. Students may also be driven invisibly to some other platform or method of cheating in those cases.
And the instructor-registration process gathers the instructors' .EDU email addresses, which could be used for other profiling, marketing, or spamming purposes (noted by Anonymous Physicist).
Arguably, even in principle, the tool does not prevent academic dishonesty, as going online to search for answers is itself already a violation. The fact that the Chegg Honor Shield turns students away without reporting them is a broken process (noted by Scott Seidman).
The Forbes article on Chegg as of January 2021 assesses:
It’s doubtful that Honor Shield will dent students’ chegging.