Squeeze-out defines any well-glued joint. Follow these tips to avoid overlooked spots of dried glue showing up during finishing.
Phot of scraping thickened glue from joint
Once the glue thickens but remains soft to the touch, remove it easily with a putty knife, scraper, or a chisel, bevel up.

What is the best way to deal with glue squeeze-out when assembling a project? I'm fed up with discovering errant glue spots and fingerprints when I apply finish.

Squeeze-out defines any well-glued joint, so every woodworker should know how to deal with it. Follow these tips to avoid overlooked spots of dried glue showing up during finishing like DNA evidence under a crime-scene black light.

Effective glue removal comes down to the timing. Wipe it off too soon and you just force it into the pores of the wood, making it more difficult to remove; but wait too long and you risk tearing out chunks of wood along with the hardened glue. Instead, wait about 30 minutes after applying clamps and check the glue. Once it congeals to a rubbery consistency, you can remove the gelled glue cleanly, without pulling up any surrounding wood fibers (photo, top). 

Photo of tape applies along joints of boxes
Apply masking tape along the joints of boxes, drawers, and small assemblies before gluing. Peel off the tape after the glue sets up.

For places where a tool can't easily reach, head off squeeze-out at the pass by masking along the joints before assembly (photo, above). Once the glue dries partially, peel away the tape and the squeeze-out along with it. For areas where masking isn't practical, finish your project parts before assembly, keeping the stain and finish off glue surfaces. PVA glue won't adhere to the finish, allowing you to pop off the dried glue with a putty knife or the edge of an old credit card. 

Photo of writing derived glue with mineral spirits
Dried glue shows up clearly when wiped with mineral spirits. The mineral spirits evaporates quickly and won't interfere with the finish.

Regardless of how diligently you remove squeeze-out, always double-check for dried glue before applying a finish. Wiping down project surfaces with mineral spirits reveals any overlooked spots of glue (photo, above). Remove them by scraping and sanding.