What kind of resistors do I need to use in this Canon remote switch, if it is possible at all with the P2N2222A transistor I have?
While perusing the IOIO board link provided in the question, I noted that the board has some 5V tolerant open-drain digital output pins.  Since the camera has built in pull-ups, as shown in the first diagram in the question, you should be able to connect the open drain pins directly to the camera.  You will not even need the external transistors.

The current sink capability of the open-drain outputs is specifed at 20mA, which easily will handle the .5 mA mentioned in the question.

In this case, the camera "expose" and "focus" would be connected to open drain outputs, and the camera ground would be connected to the controller board ground.

edited to add:

I now note that the page you cited in the question, http://www.covingtoninnovations.com/dslr/CanonRelease.html, advises adding an optoisolator to the circuit if the camera will be powered by an AC adapter rather than the camera battery, as shown in the second diagram in the "Serial-Port Cable" section.  That is probably a good idea.  In that case, you could connect the optoisolator to an output pin on your controller board that is not an open drain pin, selecting a dropping resistor to set an appropriate drive current for the diode.