Most of the other answers in this thread focus on the mathematics. This is appropriate, as this is a Q&A site for mathematics educators. However, I suspect that the question being answered ("Should I teach the squeeze theorem?") has already been addressed here many times. The distinguishing question here seems to be "Should I, as a tutor, teach a tutee the squeeze theorem when the primary instructor has omitted it?" While this question might be more appropriate for the Academia SE, it doesn't seem off-topic here.
What is your role?
Typically, the role of a tutor is to provide supplemental coaching to a student who is taking a course from a primary instructor. It is the job of the primary instructor to establish the curriculum, the grading schemata, the pace, etc. The job of the tutor is to review the material presented by the primary instructor in order to prepare the student for assessments.
Are you doing harm by omitting the squeeze theorem?
As a tutor, you serve your tutees by preparing them to perform well on the assessments provided by their primary instructor. When you help them with material that is likely to be on those assessments, you are doing your job. If you spend time on material that is not likely to be on those assessments, you are, perhaps, helping the student to learn more, but may be harming them in the sense that they may perform less well on assessments. Coving extra material is, likely, neutral at best.
Is the student being harmed by the omission of the squeeze theorem?
In my opinion, yes.
As has been pointed out in the comments and in other answers, the squeeze theorem is a fundamental result in analysis (or, perhaps more foundational, is the result that $f(x) \le g(x)$ implies that $\lim f(x) \le \lim g(x)$). Skipping this result does a disservice to the mathematics, and has the potential to create a kind of "cargo cult" version of mathematics.
Thus, in my opinion, if an instructor chooses to omit the squeeze theorem, they are doing a disservice to their students. But that is on the instructor, not you.
What should you do?
Again, this is probably a matter of opinion, but: your role is to prepare your tutees for the assessments their instructor is likely to prepare. This instructor has skipped the squeeze theorem, hence you should probably also skip it, or spend only minimal time on it.
You also need to be careful to respect the authority of the instructor. You and the instructor should appear to be a united team in front of the students. Be careful not to criticize the instructor in earshot of your tutees. Doing so only degrades the relationship between the instructor and their students, which is likely to make the overall learning environment worse.
On the other hand, it might be worth talking to the instructor. I can imagine many possible results, including:
- Students are often unreliable reporters when it comes to describing what has been taught. Perhaps the instructor will tell you that they did spend time on the squeeze theorem, and the student has simply forgotten (in which case, you really need to spend some time on it).
- Perhaps the instructor had a really good reason for omitting the squeeze theorem. I have difficulty imagining what that reason is, but you might find that they have a persuasive argument.
- Maybe the instructor just forgot about it (e.g. if they are teaching multiple sections of the course, they may have gotten confused about what was said to one group of students vs another). They may welcome the feedback, and take it as an opportunity to fill a gap.
- Or the instructor could be a real a-hole who doesn't really know what they are doing, skipped the theorem because it is hard and students often struggle with it and they want to make sure that their teaching evaluations are good. That would suck, but such is life.
Whatever the case, having a line of communication between instructor and tutor can be valuable. If it is possible for you to talk to this instructor, I would advise you to do so.