Chris --- Looks like you were right on target with the breath suggestion. Now I've been trying to take my breaths through the corners of my mouth, instead of opening my mouth every 20 seconds or so (and having to reset the embouchure each time).
Last night I jammed with three other fellows (guitar, bass & drums) on "Summertime" and "All of Me" --- and not ONE squeak!! So I must be on the right track. Now I just have to get that air intake programmed into my brain as a habit, so I don't have to think about it.