Keyboard emulation of mouse buttons

Doubtful, but as I don't have a 3-button mouse, I can't tell.

Try System Preferences > Universal Access > Mouse.
Select the "Click Option key 5 times ..." selection and exit.

Then click the Option key 5 times. You'll hear 5 quick click sounds, and the numeric pad on your keyboard will now emulate a mouse. The "5" key is a left-click. There may be one that will emulate a 3rd button.

Click the Option key 5 times again to go back to normal.

I am using a program that requires a three button mouse. Is there a way that I can emulate the three buttons on the keyboard?

Is this a native Mac program, or is this something running under X11?

If it is a native Mac program, I would not expect it to demand a 3 button mouse. It it does, then get a 3 button mouse.

If X11, then read the X11 -> Preferences -> Input
[X] Emulate 3 button mouse (then read the fine print)