I think you are correct, @william_oneb. I am going to have to use some jQuery. I found something similar to this, but I added a bit to remove the link title, so it does not show as a tool tip. The jQuery I have is a little different but it does the job.
The only issue I have is that IE still shows the URL in the lower left when you hover over it, but from what I have read, that is browser functionality that we cannot control. This may be the best I can do without hacking the main BP Core file.
Thanks for the suggestion, @william_oneb. I have added that to my CSS. It has worked for Firefox and Chrome, but with Internet Explorer it is still a clickable link, just with the default cursor and no hover actions.
This is a good start, but I would like to disable it across the board.