Hola,
In looking at this file for you (USA Counties) from ArcGIS Online, and downloading it. I found where the reason you are not able to add a new field, is because this file is a SDC file. Which is a compressed uneditable file, and if you are looking to making edits or changes to it. My suggestion would be to make a new file geodatabase, and then select out an area of interest from the USA Counties file, and then export these selected records to a new smaller dataset into the newly created file geodatabase. Then you will be able to make edit or changes, such as adding a new field to this new layer. Gracias!
SDC feature class
A highly compressed, read-only data structure that can store spatial geometry (points, lines, and polygons), and attribute data.
