To answer the original question, when I processed my first roll of black and white last year I followed the Ilford publication Processing Your First Black and White Film. From Step 15:
To remove any excess water carefully run
squeegee tongs or a clean piece of chamois cloth
down the length of the film. (Take care as any grit
caught up here will scratch the whole film.)
Now obviously this thread shows that people have formed their own opinions on the matter but certainly if you are new to B&W you're probably going to listen to the guys at Ilford who might know a thing or two about a thing or two.
I used to keep the squeegee floating in my temperature water bath until it was time to use it and it never seemed to be a problem before hanging film in my bathtub to dry. Once I set up a small darkroom I stopped bothering with the squeegee step.