This will probably be old news to some, but it's new to me...
I got a woodworking catalogue today and noticed for the first time mention of Titebond II Fluorescent Wood Glue. This is designed so that the glued joint can be inspected under Black Light after clean-up, and traces of remaining glue will become visible.
Fluorescent version of Titebond II that provides a strong initial tack and fast speed of set to reduce
clamp time. It also develops bonds stronger than wood, offers excellent sandability and is unaffected
by finishes. It contains a dye that, when viewed under a black light, enables woodworkers to inspect
the glue line and assist in the cleanup process. It is ideal for most porous materials, is easy to use and
cleans up with water.
This led me to a page that Titebond has on their website about wood glue and stains
where they give tips on removing squeeze-out.