The checkered eagle comes from the arms of the Conti di Segni family, the family of all three of those popes. It doesn't seem to have any greater meaning beyond looking cool and being visually distinctive. It continued to be used even as late as 1721, when the last Pope of the Conti line, Innocent XIII, added it to his papal arms (by this point the family's arms included a crown on top of the eagle, and Innocent XIII's arms had the more modern three tiered papal tiara).