It doesn't look like the hospital closure layer is available directly from arcgis.com. The data doesn't appear to be proprietary, just not conveniently available. With a bit of effort, you should be able to obtain the data. (Note: I am not a lawyer, and I haven't looked closely into the copyright status of this data. You should look into that yourself if you plan to redistribute the data.)

There's a downloadable table (in pdf format) of the hospital closures on that map available [here][1]. The table includes City and State data, but not lat/long or street addresses. So the challenge is getting this data in geospatial format.

- If generalized locations are acceptable for your purpose, you can join this table with a city layer.

- The street address and lat/long of each hospital are available individually in the form of a popup when you click on the hospital point. It might be possible to scrape this information from the website. Or you can manually copy and paste into a spreadsheet (it's only 109 data points, so this might be faster than learning to scrape data).

- Or you can use a geocoding service to get addresses and lat/long for the hospitals based on the hospital names.

- Or you can contact the [Cecil G Sheps Center for Health Services Research][2] directly and request the data. This would also be a good place to get an answer about whether and how you're allowed to re-use the data.

- Or you can do a nearest-neighbor join to a geospatial dataset of hospitals, eg 

  - [Hospitals on Data.gov from the Department of Homeland Security][3]
  - [Rural Hospitals Feature Layer by EsriMedia][4].


  [1]: http://www.shepscenter.unc.edu/download/11619/
  [2]: https://www.shepscenter.unc.edu/
  [3]: https://catalog.data.gov/dataset/hospitals-dcdfc
  [4]: https://www.arcgis.com/home/item.html?id=7ea850a66abd4fc1b62d47f4dabfac93