If this is possible, can a button be programmed on the phone (VVX400) to handle this operation?
It depends on your exact needs but here a few scenarios I have used:
If you have a main ring group setup that then forwards your calls either into the group or your auto attendants setup a call coverage for it's Override. In the Call Coverage for Override Mode chose Ring Extension This Many times, set it to 1. Then Ring External Number and put in the number the call should forward too. To control manually from button press set a speed call on a button to dial *20X X=the mode of chosen above i.e. *207 would set the system to Override mode. Then create a second button for speed call *200 to go back to default mode.
If you always use an AutoAttendant another way is to configure an Auto-Attendant for Override mode have it play a thank you for calling message to leave a message press 1 now to speak to our answering service press 2. Set option 2 to forward an internal extension not currently used, then set that extension to always forward to external number, sometimes depending on settings the Auto Attendant can't send calls directly to the public trunk this bypasses that issue (called hair-pinned route usually). Also if sending a call to something like an answering service you can over-ride caller id for the voice user to something your Answering Service can use to identify the call for proper answering.