Practice Your Squeaks!
Before I convince you that you need to practice your squeaks. Let's address the elephant in the room. What is a squeak anyway?
At it's most basic level, a squeak is just a high note that comes out when you don't want it to.
Practicing your squeaks can help you in the following areas:
Improve your voicing (basically, the shape of your embouchure, oral cavity, and tongue position)
Help you not squeak when you don't want to
Improve your tone
Improve your instrument's response in different registers of the clarinet
Improve your air
There are a couple of different ways of practicing your squeaks. The most basic is to play a single note on the clarinet. We'll start with Middle C. Then you want to play Clarion G without using the register key. Then see if you can get an altissimo E out. And once you have that down go for the A above that. See the example below:
The catch is you want to be able to get all of the squeaks out without using the register key. Additionally, if you are trying to reach the high E & A, you want to do this without lifting the first finger in your left hand, as we typically do.
Don't despair if it is hard for you. It can be really challenging for some people, but keep at it. In addition to practice, there are two things that can be really helpful:
Use the register key or the vent (the first finger on your left hand) only temporarily. For example, if you are having a hard time getting the G to speak without the register key. I recommend playing the G with the register key very briefly, then releasing it and maintaining that pitch.
It is also very helpful to hear the pitch you are trying to get out while you are trying this exercise. If you have a friend who plays the clarinet, have them play the note that is difficult for you to get out and try to match it. If that isn't an option for you, a drone (like the one a tuner produces) might help as well.
Once you've got your squeaks down, you can practice these concepts in a lot of different contexts. For example, in the famous excerpt from the first movement of Beethoven's Symphony No. 6, the high Ds are notoriously difficult. Try practicing the passage without the register key and see what happens when you add it back in!