In the Air Force, you have to apply but don't have to accept if selected for the residency. At least this was the case in 2013. Maybe it has changed but I can't imagine any previously signed contracts could be changed at this point.
I got my second choice after an AEGD (My 1-3 were all my top choice because they were all overseas but for the sake of being completely honest as far as the list goes). 4/8 residents in my group got top 5 or 6 bases. 2 volunteered for Korea to try and get a follow on. 1 got a base not even in his top 20. One initially got his top choice but had a last minute change to a less desirable base.
In the Air Force, you can not be stationed overseas as your first assignment without an AEGD. You will not be allowed to do many of the things that AEGD graduates will do (At least until you can become credentialed. This CAN be a complicated process). Credentialed just means that you have proven to someone in the Air Force/Chain of command that you can reliably and with a consistent level of competence that you can do a specified procedure. You will have to do this in a supervised process which can be much like dental school depending on the base and leadership. Some of these procedures may include oral and IV sedations, 3rds, Simple and complex endo, Retreats, perio surgery, multi unit pros, restoring posterior implants. Even if you left dental school with a moderate level of experience with these things, you still wont be able to do them in the Air Force without going through the credentialing process. Doing an AEGD allows you to become credentialed in some, if not all of the above. Also, if you don't do an AEGD, more than 80% of everyone else around you will have more training than you and will automatically look better than you when it comes to special assignments, awards, advancement, etc. Whether it is a good thing or not, being a Vanilla just doesn't look as good. I know each service may vary in this process. An AEGD has a whole list of pros and cons but I just wanted to give a taste of what it means to your practice of dentistry within the Air Force.
There are so many factors that are involved in handing out assignments. No one really knows all the details except for the people that work as the assignment officers. Recommendations from your residency director, Stratifications from residency instructors, competency in complex procedures like endo and 3rds, how much military bearing you exhibit, how much you drive others crazy, can you hold your own without a specialist down the hall, how many spots are promised as follow-ons for people who went to undesirable locations voluntarily, how many spots are available of O-3s. I am sure I haven't even scratched the surface. And when it is all said and done, even if you are God's gift to dentistry and the Air Force/Navy/Army, you could still end up in Minot ND, Cannon NM, or Afghanistan if the service needed you there.
I thoroughly enjoyed my AEGD and am glad I did it. I gained a ton of experience in things I never was allowed to touch in dental school. It allowed me to have an amazing assignment at an awesome base overseas. Even if you plan on getting out after your first assignment, I think it is a good return on your investment of time.
Sorry so lengthy and maybe a little off topic.