I have never - ever - been tempted to stain first. I don't understand why one might.
Squeeze out should not be an issue unless one tries to remove the glue while it is still liquid. That's easily the worst thing to do.
I know what the titebond instructions say and it is nonsense. Never try to remove squeeze out with a wet rag. Never try to scrape it off while it's wet. Leave it be.
Deal with squeeze out by leaving it alone.
Let it dry, sitting lightly on the topmost fibers, do not touch it, and when it is mostly hard with no no liquid core it'll pop right off like it was never there. If you let it harden completely it'll be a little tougher to get off. It'll still come off.
This applies to all titebond, PVA, elmers, white glues, hide, and epoxies too.