Hi Leigti!
I'm so sorry to hear about your chicken and the situation you are faced with.
I have used our pellet gun to put down a chicken. It is effective so I would think that a BB gun would also be as effective. We use metal pellets with our gun.
*Contains graphic depiction for those who may not wish to read the following.
The best way I've found to dispatch the majority of birds and animals with a gun; a pellet to a .22 rifle...is to place the gun at the base of the skull at the back of the head and have the pellet/bullet pointed high enough into the skull that it's close to eye level or in that area. It seems to be the quickest method resulting in the least action and blood from the animal/bird from any method I've used. The bullet/pellet severs the spinal cord and they may spread their wings out/jerk slightly/move their legs and it's done quite quickly. The only thing you don't want to do is not have the BB pointed too low so it aims at the bottom of the face or jaw. The pellet to the brain kills them very quickly so they won't be feeling anything. The movement is the resulting nerves.
This is by far my prefered method. It's nice because they can be resting and not expect anything. It's also nice that you don't have to look them in the face. I also find that having to point at the head from the face produces more blood.
Just make sure you have extra BBs on hand in case it does require another BB if the placement wasn't quite perfect.
I hope this wasn't too graphic of an explanation for you...but I wanted to be thorough to prepare you for what to expect.
Please know that if she is sick enough that you feel it necessary to put her down...that you are being very kind in not letting her suffer. This is very commendable. I'm so sorry that you have to go through this.