All good suggestions. We use Short Cut. In other studios, I have seen the following configurations; one of these will work well for you.
One station uses the Tascam MD-350 minidisc, which allows editing. It works well for phoners.
http://www.tascam.com/products/md-350.html
Another professional configuration I've seen is a desktop with Audacity (free) loaded. (Actually, the desktop has Audacity and EBRCart machine software). The Audacity program is hotkey configured so that F1 = play, F2 = record, etc. Now, this particular installation went a step further by eliminating the keyboard and, instead, uses the X-Keys product. Each function is assigned to a key. Press a key for "play", another for "stop", and another for "record", and more. The keys are labeled as such. The installation is very clean and professional.
http://www.xkeys.com/xkeys/xkdesk.php
You could certainly fire the play function from a board closure. One way would be to use closures off the "printer" port. The closure could start the playback. You'll need a simply program to configure this.
A more advanced way would be the Measurement Computing interface. It accepts various closures and sends commands to the computer. This interface may certainly be over-kill
http://www.mccdaq.com/usb-data-acquisition/USB-1208LS .aspx
You could add a button box as well as board closures to control the functions of Audacity.
Let us know how the project turns out.