Because these countries are going to come. In you look at Turkey, yes, they are turning towards Europe, but they have a huge Kurdish domestic population and view an independent Kurdish state as a threat to their national security. If Europe punishes them for this, they simply turn towards Asia and away from Europe, especially with China and Russia dying to increase their influence in the region. Turkey can afford to do this.
Iran is not Arab no. But they back the Shiites in Iraq to a huge extent, and do not want to see Saudi Arabian and Sunni influence increase in the region either. Iran is already heavily involved, and will continue to stay involved to thwart any attempts and Saudi Arabian influence coming in.
Saudi Arabia is in the same situation. They are competing with Iran for regional hegemony. To idly sit by and let Iran increase its influence and dominance in the region is simply unacceptable, and they have said exactly that.
Syria is not the problem, I didnt even mention them because they will be mostly a non-factor in a post-war Iraq or a post-US Iraq.
It doesnt matter that we give aid to Saudi Arabia or Iran is not Arab or Turkey is trying to get in the EU. What matters is how these people view the security in their region. None of these states will trust the other one and all will be involved in a power struggle for Iraq after the US pulls out.
They are not going to outright invade, it will come in the form of "peacekeeping" after the US leaves, and will be played up as them cleaning up the mess the US made.