getting yourself AFU by riding on a broken/improperly repaired binding. Nevertheless, if you are determined to try to "glue" it please check out the following link and learn how to properly repair plastics.
http://www.urethanesupply.com/howtorepair.html
I recommend you do additional research as well if you plan to do this.
The Burton binding you are referring to, to the best of my knowledge, has a Polycarbonate baseplate and Nylon highback. It sounds like the part you are referring to is the (non-adjustable) heelcup which is part of the baseplate on this one. Polycarbonate is a thermoplastic and you should be able to "weld" it back together and without worrying about the integrity of the binding afterwards. However, this is largely dependent on skill and proper tools & materials. Ensure that you can correctly identify what type of plastic you are trying to repair and follow the methods outlined in the link above. Also note that the methods outlined are intended for repairing plastics on automobiles. And of course, you`d be best off consulting with your local shop or calling around to find a professional to do it right. Be prepared to buy a new set of bindings because if there currently is an effective warranty you will definitely void it.