Saxophones are fun and comparatively easy instruments to learn. There are some inherent difficulties that most beginners face however and these difficulties can be addressed with a few short and easy steps to assure the best possible path to success.
Stop blowing so hard. More often than not a sax that squeaks continuously is the result of far too much air. Tighten your embouchure, which is the technique of your lips, face muscles, and how it affects the mouth piece set-up. Now focus the air stream to see if this fixes the problem. If the sound feels like it is jumping from one octave to another it is probably a leak.
Check your reed. Playing a reed that is too hard in strength (#3 to #5) might be far too hard to blow. A hard reed requires more air to vibrate and uncontrolled air from a beginner can cause constant squeaking. Try a reed strength of #2 to #2 ½.
Change reed brand. Reeds are cut from cane and therefore grown. Certain crops can be affected by many elements in the soil and weather, as well as the harvest and processing. A particular brand of reed could have a bad batch or is just cut in a way that is problematic to you. After changing strengths, if the squeak continues, try an assortment of different brands. Some examples are Vandoren, Rico, La Voz and Hemke.
Check your mouthpiece Stock mouthpieces usually have a close tip opening and a small closed chamber, which forces air into a small space. Too much air can cause back pressure and cause squeaking. Look for an open chamber mouth piece.
Check to see if your horn has a leak. To do this you can see if light comes through when the keys are closed, but most times you will not be able to see this without a leak light. A horn that squeaks most often has a leak in the register key. This is the mechanism on the top of the neck of the horn. A competent technician will need to be enlisted to identify and fix this problem.
Go through this process in order. Don't purchase anything until you know that the squeak is not caused by a playing problem.