The package being tampered with was problematic, but the hole is made when the container is filled with blood not when removing the blood. If you're removing blood, you just take the lid off.
To be fair, that wasn't the only scene where I felt that way ...
Hard to find good pictures, but you can see here: http://i.imgur.com/mhdkpui.jpg some vacutainers (newer, but they work the same) that have been filled, complete with puncture holes. That's just how you fill them.
It's a lot more compelling that the evidence box has been tampered with, in my opinion. You can find a few phlebotomists confirming this in the makingamurderer subreddit, if you look around.