Before anything else, check out your reeds. On mouthpieces with large tip openings, a reed that is warped and not sealing on the table of the mouthpiece will squeek. A good check to see if the reed is sealing is to 1. take the neck off with 2. Cover the tenon side of the neck with your palm to make a seal 3. Suck air through your mouthpiece, the read should adhere to the tip opening-forming a seal. When you remove your palm from the other part of the neck, the seal will be gone, causing the reed to make a popping noise when the seal breaks.
In addition, I have had difficulties with certain brands, Alexander D.C.s are the worst in this regard, that chirp and squeek. I have had the most success with Rico Jazz Selects.
If reeds aren't the issue, check your embouchure. Are the corners of your mouth in an "oh" shape as if you going to whistle? This shape prevents biting and too much pressure on the reed.
Finally, rather than the chamber of the mouthpiece, the shape of the tip opening and the rails is more likely a culprit.