I'm not sure if any of the following applies to other states, but here in NJ the employer can ask you if you have ever filed a WC claim & the prospective company can have their insurance company perform a check to see if you have. In addition, if you're a healthcare worker licensed by the state (LPN, RN & CNA) it is mandatory that you have a physical before beginning work (That, I would assume is the same in all states) & can also ask you when you have the physical if you have had a previous WC claim, or any prior injuries/pre-existing conditions etc,. All legal.
While I agree, it is an invasion of privacy of sorts, it is also a way for employers to protect themselves & control the cost of insurance. Unfortunately, there are many who "make a career" out of filing WC claims. And if it applies-they are also able to receive a payout of they are out on WC for more than six months. Meaning, lets say someone injures their leg & after 6 months or so they claim they are still unable to work; rather than the insurance company continuing to pay them for lost wages, they assign a dollar amount-Your arm may be worth 15 grand, a knee 20 grand, etc., so they pay them that lump sum of money. That & more equates to a large expense to the employer. I've seen quite a bit of the above personally. I'm not saying that applies to the current situation of course-only that's why employers may feel they have to do it.
In addition, here in NJ, WC claims are considered a "no fault" system. You could lie & still receive financial retribution & treatment. Your employer/insurance co. could lie, refuse treatment etc.,but if you file a complaint with the WC court & your claim is even vaguely legit, the employer & their insurance company have to comply.
With all that being said & I'm certainly not an employment attorney, but I cant imagine a bite on the hand without any lost time would prevent another facility from hiring you.