Neither VoiceAttack, nor any piece of software interfacing with the Windows joystick APIs, can press buttons on an existing joystick device.
The Windows joystick APIs (I.E. the means Windows provides to software to interface with joystick devices), unlike the keyboard and mouse APIs, are not designed to allow for this functionality.
Again, this is not a limitation of VoiceAttack itself. Microsoft chose to design the APIs this way.
If it doesn't need to be an existing joystick device, this limitation can be worked around using a custom driver that presents itself to Windows as a joystick device, but is explicitly designed to take instructions from software.
This will be a completely separate device.
There is
a VoiceAttack plugin available to control said driver.