How do I fix this?
It looks like the top faces of your cylinders and the top face of your box are at exactly the same height. Moving one or the other up or down a bit should fix it.
I want it to be at the same height. Is there no other way?
duplicate cylinders and scale to poke through the box and then boolean the box with them so they cut out the cylinder shape.
Why?
What is your end goal?
If you built up physical imposable objects then ou have to deal with some possible problems. 