The governor of Illinois is not limited to veto or sign; he can selectively veto with recommendations for a law he will sign, and that puts the ball back in the Legislature's court to approve the changes or override the veto. There is some discussion as to whether he might gut and replace with the "Chicago" bill that was voted down (giving Chicago home rule).
Of course, the whole thing has a hidden political calculus that has more to do with who will be then next governor of Illinois, Quinn or Madigan. Whoever blocks concealed carry in the state will almost automatically lose the vote of the Southern Illinois democrats. The tricky thing is that they both want to block CC. but neither wants to be responsible for it. Quinn is trying to force Madigan to file for cert (and seek a stay of the appellate decision and injunction pending the cert app), while Madigan is trying to wait Quinn out on his response to the bill that is in front of him. Whoever acts first loses.
Right now, it seems that Madigan has the upper hand; the time on the extension to pass a bill or face an injunction runs out before the time to file the petition for cert. Technically, however, Quinn gets 60 days to respond to a bill on his desk, and that time runs out after Madigan's extension--but after the injunction is set to issue. Will he risk constitutional carry to force Madigan to file?