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. 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 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 get a geospatial dataset of hospitals (eg https://catalog.data.gov/dataset/hospitals-dcdfc) and do a nearestnearest-neighbor join. to a geospatial dataset of hospitals, eg