The problem with your classical analogy for entanglement is that it doesn't match the data. Or rather, it only matches the data for quantum properties that are similarly blue or red.
The non-classical properties of entanglement start appearing once you start measuring combinations of the redness and blueness of those balls.
Let's say that instead of looking at the balls, you pass them through some machine that will let a red ball pass through with some probability P that you control; if the ball is blue, the machine will let it pass with probability 1-P. Let's say further that you have three such machines. You set the first machine to P=1. You pass each ball falling from this machine through a second machine, which has P = 0. You will never see a ball pass through to the end - if it were red, it would pass the first machine, but not the second; if it were blue, it would not pass the first machine at all.
But, let's say you now put a third machine between the other two, and you set P = 0.5. With classical balls, nothing changes - a blue ball doesn't make it past the first machine, while a red ball goes through the first, may or may not pass the second, and never makes it through the third regardless.
However, a quantum ball actually has a chance to pass through the 3 machines if you set it up this way. In fact, that chance is pretty large - more than half of the balls will start passing once you add the middle filter machine.
Still, this is easy to explain if we assume that the middle machine actually paints the ball instead of just detecting its color. This is where the entanglement experiment comes in: if you pass the pair of balls through the three machines, with ball 1 passing through machines P=1 and P=0.5, and ball 2 passing through P=1, you will find that sometimes both balls make it through, even though both balls can't be red at the same time, and they can't communicate about passing through the P=0.5 machine (you can repeat the experiment with the balls being taken arbitrarily far away before passing through the filters).