Hi Alan

"La que está sentada en la mesa de la esquina se llama Verónica".

Generally, the word "esquina" is used for the outside corner of an object, e.g la esquina de la mesa, or you could also say "Mi casa está en la esquina" (My house is on the corner [of the street]), while rincón does't refer to the "shape" of a corner. Rincón is more like a small area inside a room or a bar for example. 

In the example you are referring to, "La que está sentada en la mesa de la esquina se llama Verónica", you need to imagine that you are looking at a few tables, say, outside a bar, and the tables are arranged in a square shape. You are pointing at one of the tables that is placed in one of the corners.

Does it make sense?