Glue squeeze out is the bane of all woodworkers. For applying backers, you need to determine just how much glue is really necessary and where it should be applied. The entire back of the piece does not need to have glue on it. I use a white glue, that dries clear, but use it sparingly. If the piece is relatively small, I use Aleene's Tacky glue. It's a little thicker than regular white glue, so you can use a toothpick to apply it, if you want and sets up pretty quick. If the piece is large, and I need more open time, I'll use Elmers White glue. Again, applying it with the smallest utensil I can, that will do the job. I've used small, cheap artist brushes with good results.
If there are really small, delicate areas that I want to apply glue, I'll save them for last and use a tiny bit of gel CA glue, just in those spots. The CA glue is easier to get pinpoint control, but it sets up very quick, so that's why I use it only where necessary and right before I'm ready to apply the backer.