NO, NO, and NO!!! These regulations, as written, push/suggest/require an employer to ask (or know or suspect!!!) personal health information. Please these consider quotes [highlights mine]

§40.A.

3. Employers shall develop and implement policies and procedures for employees to report when they have tested positive for anti-SARS-CoV-2 antibodies through serologic testing….

4. Employers shall develop and implement policies and procedures for employees to report when they are experiencing symptoms consistent with COVID-19. Such employees shall be designated by the employer as “suspected COVID-19”.

5. Employers shall not permit known COVID-19 or suspected COVID-19 employees or other persons to report to or be allowed to remain at work or on a job site until cleared for return to work or the job site (see §40.B).

You can’t do this, you just can’t. It flies in the face of everything the HIPAA stands for.

The regulations list HIPAA in the definitions but only give them a token reference in the duties of creating “the preparedness and response plan.” [§70 C.2.B. To the extent permitted by law, including HIPAA, employees’ individual risk factors (e.g., older age; presence of chronic medical conditions, including immuno-compromising conditions; pregnancy; etc.)].

STOP making employers ask very personal questions about the health of their employees. All they can be required to say is if you’re sick, stay home. If you’re sick for more than a few days, at best, have your doctor write you a note. NO ONE SHOULD BE REQUIRED TO ASK FOR or PROVIDE A SPECIFIC DIAGNOSIS be it Covid-19 or otherwise.

Now, would I personally tell my employer (and certainly my co-workers)? Yes, absolutely. But that’s me. You’re The Government. You cannot compel me to tell anyone what’s my current medical status.

Encourage all you want. Just change the “shall” to “should” and we all might just do as suggested and go along.