I'd like to add my own button link next to these two. Is this possible?
Not sure why this question was dismissed so quickly. It's actually a valid question, and would be really useful to be able to add a button up there, or even edit the content of each screen option.
I know this is an old question, but...
If you want it to just add a button up there that would link somewhere else. You could just add some jQuery to add the button.
Now if you actually want that button to work in the same way as the other buttons, you would need to dig into the admin files.
The file you are looking for is:
The complete code could look like this:
Those two tabs will be converted into a menu in the admin bar in WP 3.3. Check out the beta.
I have written a simple and exact clean solution for adding your own such button:
Add css instructions somewhere: