Once torn asunder, it is not clear that these pieces can, or even should, be put back together in Iraq, just as they were not in Yugoslavia. Many of the Christians have left the country, with many arriving in the US after the passage of the Refugee Crisis in Iraq Act was passed in early 2008. But the Sunnis (including the militant group ISIS) appear to be supported by Saudis, the Kurds are backed by Turkey, and the Shia are supported by Iran. Indeed, the Iraqi prime minister, al-Maliki, was in exile in Tehran and Damascus for most the of the time that Saddam Hussein (a Sunni Muslim) was in power.
In the meantime, it appears that most of the refugees who fled Mosul--a predominantly Sunni Muslim city--when ISIS took over headed to Iraq's Kurdistan region, since that was relatively close and populated by Sunnis. It now appears that many of them are headed back to Mosul, convinced that life under the Sunni militants will be OK. Oddly enough, it seems that the breakup of Iraq, if it happens, will make more sense than the current mess in Syria. Only time will tell.