I can not think of a way to accomplish this off the top of my head as polygon symbology is continuous. Plus, you are wanting the crosses to start and end at each corner. This doesn't sound possible especially when you have polygons of varying sizes in the map. I haven't looked completely through all of the options but I believe this cannot be done using symbology. Polygon symbology, unlike point symbology, is continuous because of all of the different types of shapes and sizes that polygons can come in. Points on the other hand stay the exact same.
Potentially, you could use code such as Python to grab the vertices of each polygon and create crossing lines to give the 'X' effect, however, these lines would be features and not symbology. This would not be ideal for when new features are created or moved.