Is it common to get cuts on your fingertips when playing the acoustic guitar?
Yes, when you play a badly set up guitar. No, when you play a well set up guitar.
Someone will no doubt mention callouses, which apparently help to stave off this sort of thing. Keep playing a badly set up guitar, and you will get them!
Squeezing too hard it will make the problem worse. If your hand position is good, you should not need to squeeze very hard. Check your technique in that regard.
Personally, I notice that dirty strings or sweaty hands can be a problem. Rubbing moist skin is more damaging that when everything is clean and dry.
There're different sizes of string, varies from light to heavy. If you've just started to learn guitar, light strings are sharp so using medium or heavy string size might help to prevent cuts on fingers.