There's a few issues. The first one is the easy one. It's mentioned by several answers already. You start by assuming you have freewill and thus prove that you have freewill. Pesky construct that one.
To get around it, we need the computer to be smarter. Much smarter than you're giving it credit for. The version you are looking at is the one where the computer prints out what would have happened had it not printed anything. As you noticed, it is rather easy to do something different, but that doesn't refute determinism because the prediction was for a hypothetical universe where nothing was printed. In this world, something was printed so the prediction is no longer 100% accurate. You might as well have asked it what the lottery numbers were in an alternate universe!
The more interesting case arises when the computer prints out what will happen, accounting for the fact that it printed something. These predictions are far more nuanced and much harder to disprove with such simple constructs. As DeeDuu mentions in their answer, there's no guarantee that you merely wanting to disprove the computer will actually lead you to succeed. In fact, the argument for the computer is that, in fact, you cannot disprove it. Trying to disprove it may simply yield unexpected results (such as breaking your leg going down the stairs trying to stop Mr. Jack Roberts from eating cereal).
The more interesting issue arises when you start playing mind games with the computer. When you start trying to play along when it thinks you'll do the opposite and so forth. You can play that game for a while, but there's some interesting mathematical oddities that arise. You see, the computer needs to predict what will happen, including what will happen within itself while the calculations are taking place. Otherwise, how can it figure out what it might print on the paper, so that it can figure out what you would do with it.
There's a theorem out there known as Godel's incompleteness theorem which starts to apply here. It turns out that the computer you seek to create cannot possibly exist within the universe. There's a strange pattern which occurs when a perfectly truthful machine tries to predict everything, including itself. It turns out that such a machine simply cannot be made from a mathematical perspective.
What you need is an Oracle: a device which can tell this sort of truth without being part of the universe at all. However, once you start adding oracles, it becomes impossible to prove their correctness -- the correctness of an oracle is always assumed, rather than proven.
Thus, the question becomes not whether the universe is deterministic, but whether such an oracle can be created outside of the universe.