Q:

Is it common to get cuts when playing the guitar?

Is it common to get cuts on your fingertips when playing the acoustic guitar?

A:

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! 

A:

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.

