I can't figure out a method or find a tool that would completely delete (or disregard) just the overlapping polygons (red) that intersect with the Puerto Rico polygon (black). I essentially want to create a shapefile with just the polygons that don't intersect with the black polygon. I am using ArcGIS 10.3.
You can remove the overlapping polygons with a Select by Location then Delete.
First use Select by Location, either with the Select Layer by Location tool or from the Selection menu in ArcMap. Select features in your red polygon layer that intersect features in your black polygon layer.
Then Delete - either with the Delete Features tool or Start Editing, and delete selected records.