The clarinet requires a lot of air, but if you use too much at once, it causes the instrument to produce an unexpected overtone, aka a squeak. Keep your air fast and steady for an even sound across all registers.
Chances are if the inner and outer embouchures are both set, have the student take in much more mouthpiece then they think they need. They should be able to take in enough mouthpiece where the low note doesn't “squeak” on its own but the high notes pops out easily.
The most common problem is the crow's foot that was mentioned. If it is out of adjustment in a way that the B key doesn't close the C key, then the B will stop working before the low E. Only a bigger leak will also stop the E from working.
Should you puff your cheeks when playing clarinet?
Clarinetists with underdeveloped embouchure muscles may puff their cheeks. They will puff their cheeks because their embouchure muscles are still not strong enough or they are tired. In order to fix puffed cheeks, encourage the students to take a break when their embouchure is sore.
The clarinet is no harder or easier than any other orchestral instrument that a beginner may learn. It is the usual case with an instrument that you blow that arguably the hardest part of learning is getting a sound out in the first place.
Slow air speed at the beginning of a note is also a common problem. When the air speed is too slow or not voiced with "ee" the sound may start with a "scoop," be fuzzy and/or unforcused. Have students practice starting notes using a regernece pitch tuner as a model for clarity. (See Louder Clear/Softer Clear Exercise).
In regards to the tonguing you are likely either letting too much of the tongue touch the reed and/or are tonguing too low on the reed. Try using a "tee" syllable and be sure the very tip of the tongue is touching just below the very tip of the reed. This should help to create a cleaner articulation sound.
Both have their easier and not so easy aspects : I found it easier to get an acceptable sound to start with on the clarinet, and also found the posture rather easier. Once you're going on the flute though, a lot of the basics will fall into place with practice (and good guidance).
On the clarinet, playing C and blowing hard produces a high G. The clarinet is the only wind instrument that can reach such high notes. With more tone holes than the recorder and an extended register, the clarinet takes full advantage of this property.
A good rule of thumb is you should replace your reed every 2-4 weeks, no matter how often you're playing your instrument. You may want to replace your reeds more frequently if you're practicing several hours each day. Some reeds also may not last as long as others, every reed plays slightly differently.
From two sounds with equal quantity of overtones, the one with less harmonics (less overtones that are integer multiples of the fundamental) will tend to be perceived as dirtier. Loudness of the overtones.