Across four-year universities in the U.S., do professor emeriti tend to receive full health benefits? Do professors ever retain health benefits in retirement? If not, how do retired professors maintain health insurance coverage?
Generally, being a "professor emeritus" just means that you keep your academic privileges such as being able to advise students, to access the library, etc -- but not the benefits of employment such as a salary, health insurance, etc.
In that latter regard, they are retired just like everyone else, and that means that they need to have a way to support their life and their health insurance independent of any previous employer. In the US, that generally means getting Social Security, drawing from previously accumulated 401(k) moneys, being on Medicare, etc. Some (previous) employers allow to "buy into" their health care plans, but you will then generally have to pay the cost of insurance yourself.
This is a contractual matter and in some places a matter of state law, though I can't cite sources on the latter. "State policy" might be a better way to say it. So, it varies. My last employment was at a private university and I still have the same health insurance, paid by the university, that I did when employed, though it now coordinates with Medicare. They actually pay Medicare premiums. But not every place is so generous. Dental isn't covered and I have very modest co-pays on visiting doctors or hospitals. That was the same back before I retired. I never need to think about cost to go to a doctor. The plan includes pharmaceuticals as well, with some medicine free and some at low (co-pay) cost.
But there are a huge number of colleges and universities in the US and, while some fall into state systems with coordinated policies, most have institution specific policies. There are not laws, national or state level (AFAIK), requiring such things, so it is impossible to make general statements without a statistical study. My guess is that it happens "often" but not universally. Probably not "typically", but you'd have to do the study.