To me, the more natural idea is squeezing the activity in, not squeezing minutes out. Some ways for that are: I'm going to try to squeeze (in) a few minutes of reading into my day. (My day is full. But I am going to try to add a few minutes of reading. I'm curious what others think about the first "in". It seems redundant but sounds ok to me. ) I'll squeeze in some reading. (I'll find some time to add reading) Another task to do? I'll squeeze it into my day if I have to. (here I wouldn't add the extra "in")

I would say “squeeze a few minutes INTO your routine for reading.” “In” isn’t quite right; it needs to be “into.” “Out” is grammatically OK, but INTO is better because it puts the emphasis on adding one thing to your routine, rather than on limiting the time you spend on other things. “Out” would be better for situations where you are trying to eliminate/lessen something. Example: “The computer programmers are trying to squeeze a few more milliseconds out of the run-time for the operation.”