Here is the problem, that even though you have health insurance, does not mean that the health insurance is going to pay your medical bills. HUH?
Most people today have high deductibles, thus until they pay out the amount of their deductible, then insurance starts paying. Thus why the reason of wanting to know if you are employed or not.
Either way, (if) you have problems paying the medical bills, then most providers will allow you to pay them on a payment plan. This maybe the (one) and only time to tell them you are unemployed (now), thus if insurance does not pay, then make arrangements to pay back what you can.
Now, anyone else, then when asked, then say (what) your profession was. When people meet, then usually each will ask what type of work they do, but most never ask if employed or not, they just assume you are working and then talk about other things,.