Ok, I know (and recommend) that people should do this in a dev environment first, but I took the plunge and just did it. Hopefully this will help someone in my position.
I went and saved the security.php file and stored it off server.
I then put my store in maintenance mode (with an IP allowance for my IP).
Then I went in and deleted the old security.php (I guess I could have renamed it instead, but I wasn't thinking about that at the time)
I went into putty and applied the 7405 patch using SSH.
It worked! No errors. Patch applied successfully.
I went back onto my site and tested user interactions that I could think of ( I may have missed a few but to me these are the most important):
Add to cart, remove from cart, apply promo code, remove promo code, use search bar, sign into account, sign out of account, checkout.
All work fine.
Took website out of maintenance mode.
Warned my coworkers who deal with customers to tell me of ANY issues (or to just come get me so I could talk to users).
This was probably not the best idea (doing it on a production server). So far, however, it seems to have worked out ok for me. I will post back here if I have any issues.
