Apparently squeaky cheese happens as a result of very, very small bubbles of gas in the cheese that cause the cheese to be textured as you bit into it. Instead of being flat and smoothly sliding along your teeth, it is rough and kind of shudders as it moves. This is what causes the noise. I can't remember where I saw that explanation, so please take it with a grain of salt.
My interpretation of that is that you probably aren't stretching the curds enough. Mozzarella and cheddar are actually made in a pretty similar way up until you stretch the mozzarella. As you rest the mozzarella before stretching, it's similar to the cheddaring process. The cheese flows while the pH is still high enough to do so and it slowly elongates the proteins. Here's a picture of Gianaclis Caldwell making cheese curds:
You can see how the curd forms those layers (which is incidentally one of the goals you are striving towards when you are cheddaring). My understanding is that it's those layers that you bite through that cause the squeak.
With your mozzarella, I think it's likely that you haven't hit the optimum pH to stretch and therefore it hasn't stretched out enough to smooth out those striations. The best video I've seen for showing how to develop the curd is actually a commercial burratta producer:
https://www.youtube.com/watch?v=SrH0mKciUKI When I've made mozzarella, I tend to stretch a bit too soon. That video really helped me understand what I'm trying to achieve.
But... good mozzarella is super tricky :-) I think it's the type of cheese you could make 100 times and still find lots of ways to improve.
