There are so many jokes (mostly bad) that I can think of about squeaking that I'm not going to put even one up here, so here goes with the serious stuff.
What is squeaking?
If you don't know, I'm not going to tell you.
Oh, heck, I might as well.
Squeaking is an unwanted jump into a very high note.
A squeak isn't something mystical, it's actually a true note on the clarinet or saxophone, a note that is one of the higher "fundamentals" of the note you're playing. A "fundamental" is the lowest pitch of any note, but there are more than one fundamental for any note, the first fundamental, the second fundamental, the third fundamental, and so on. No single note on the clarinet has only one fundamental. If it did, it would be a pure tone and it certainly wouldn't sound like a clarinet, it would sound more like a flute or whistle.
There's a lot of acoustics theory in that, but even if I understood it all, you wouldn't be interested. All you want to do is not squeak!
The important thing to remember is that a squeak is a jump to another fundamental of of a single note, the note you're trying to play--a "legitimate" note, not some acoustical abberation. It's natural and it's "real", but when you want to play a given pitch and it comes out 'way higher what you wanted you're not happy, and damn the physics of it!
So let's move on.
Why do I squeak?
In order of probablity, and in any combination of more than one:
Your fingers aren't covering the tone holes (the main reason why beginners squeak).
But you'd be surprised how often this happens, even beyond the beginner stage.
Even after several glacial epochs of playing, as I was practicing for a recital, I suddenly developed the squeaks in several notes in the chalumeau register (from the 3-finger left hand C to the thumb and 1st finger E ) and from the clarion G to clarion B (same fingerings, but with the register key added).
It was so frustrating and I couldn't figure out why I was doing that that I got with a pro player friend of mine. I played an F major scale for him, one octave, and he immediately saw the problem: Inadvertendly I'd started playing "flat-fingered" instead of with my fingers curved, and this caused me to let the tone holes leak a little, causing the squeaking. I had managed to stray away from proper finger position because of the arthritis in my left hand!
Your reed or your embouchure is seriously unbalanced
An unbalanced reed will squeak easily because one side of the reed (the thinner side) will vibrate much fast than the thicker side (if the thicker side vibrates at all.
Change your reed and see if it still happens! (And more important, learn how to balance reeds.)
If you're playing with a balanced reed and everything else in this list is okay, you're almost certainly favoring one side of your embouchure over another.
This produces the same effect as an unbalanced reed: you're damping one side of the reed while letting the other side vibrate more freely. (Shame on you for not forming a balanced embouchure!
Leaking pads, especially in the upper joint of the instrument
Find the note or notes that squeak frequently and look at the pads on keys above those notes. Are they frayed, loose, or damaged? Has the key been bent so that, even though the pad looks good, there's a gap between the pad and the tone hole?
There are a few ways to test for leading pads (more reliable than visual inspection):
Try to play the next note below the notes that squeak, and while you are playing, have a friend press down on the pad cups for all the tone holes that are down above the lower note. Although it's possible that more than one pad isn't sealing, if there's a leak, when the pad cup is pushed down, the squeak will likely go away and you'll be playing the note you fingered.
If you find the leaking pad, a quick fix is to wrap some food wrap or teflon tape over the pad. This works best for small leaks, but can be a real life-saver in a pinch.
Plug up the lover end of each joint, seal the upper end with your mouth, and blow with moderate pressure. If there's a leak, you'll probably not only feel it, but hear it.
Old timer's like me who smoke would blow smoke into the joint, and would therefore be able to clearly see if and where there were leaks. (The clarinet did not die from secondhand smoke.)
Weak or missing springs
Do the "breath test" in #3, but use a bit more breath pressure. If a spring is weak or missing, the pad will open just from the pressure alone.
If you find a weak or missing spring, wrap a rubber band around the key to hold it down. (This saved my butt twice in the past.)
Loose tenons.
If any of the joints on your clarinet wobble or part of the cork on a tenon is missing, there may be a leak that causes the squeak. (This is especially true with the mouthpiece/barrel/first joint assembly.)
Wrapping teflon tape around the tenon to tighten it up works very well.
Cracks!
Cracks anywhere in a joint or barrel can cause leaks, but it's very unusual for a crack in the body to cause a leak and cracks in the bell don't at all.
The really bad cracks are those that go through the tone hole "chimneys" (the raised part of the tone hole) of tone holes or that go through the register key hole.
Sadly, there is no emergency fix for these; cracks have to be pinned or sealed, and only really good technicians can do that.