as far as i can make quillard is a type of sail boat and can also refer to a type of race contestant. Could it be that "quillard" are those who are leaving for their assignment or race (so to speak) as in "coureur" as opposed to "Recrues" who are newly arrived.

These appear to be those who are about to be "bowled over" (the way bowling pins are bowled over), that is, those who are finishing up their tour of duty and about to return to civilian life.

Here is a quote:

Une quarantaine d'officiers et sous-officiers de réserve ont fait leurs adieux aux armes dimanche matin à Belfort.

Au cours d'une cérémonie exceptionnelle au 35e RI, peut-être la dernière du genre, le général Bachelet commandant la 7e division blindée et le CMD de Besançon s'est adressé aux réservistes-quillards, atteints par la limite d'âge en des termes élogieux : «
Vous êtes une étroite complémentarité entre les cadres de réserve et d'active.
Je vous suis reconnaissant envers vous tous qui avez pris une part active dans
les armées de réserve ».
http://www.lepays.net/jdj/97/04/08/RP/article_22.html

Whoops. I meant to write "ending," not "ended"

...who've been through basic training give their letterbox to the new recruits

Whilst what immediately springs to mind is the nautical term "keel boat", that is obviously not approriate herre. The Nouveau Petit Robert gives the following explanation :

"n. m., mil. XXe argot militaire : un soldat qui est sur le point d'être libéré du service militaire".

That's the meaning. Now to the translation.

Unless there is a very specific English term (bearing in mind that military service does not exist in the UK), then two contrasting terms for the gusy who are about to leave versus the new recruits is what you need. But then you probably knew that.

'fêter la quille' is a colloquial expression used in France to refer to the end of the period of compulsory military service.

A 'quillard' is therefore a young soldier who is finishing his 10 months of military service and returning to normal (civilian) life.