Do you have time to squeeze me into your schedule? I would like to have more time to squeeze in my hobbies. If you want to chat more I have availability in my schedule today.

I'd say "I'll squeeze out some time to read books". Just so you'll understand the metaphor, the thing that's getting squeezed is not the time during which you read books. It's the time that you've allocated to your other projects that will get squeezed. You're going to allocate less time to those other projects in order to make room for books. In other words, you're squeezing time "out" of the other projects.

Your example is close. You forgot “in”. I squeeze IN some time to read books (when I am not at work). For an appointment: The doctor is very busy today but I can squeeze you in at 3:20.

Yeah, your sentence is correct. “squeeze time to do something” is a common use.