|No, not that kind of engagement|
"Engagement is more than job satisfaction. It involves employees being intellectually and emotionally connected with their organizations and work teams because they are able to do what they’re best at, they like what they do at work, and they have someone who cares about their development at work."
And it turns out that it doesn't matter much whether you went to a prestigious public or private college/university. You are about equally likely to be in the lucky 39% regardless of what kind of college/university you went to. The only exception is for-profit colleges which don't do so well by this measure.
You can find the poll at http://www.gallup.com/file/strategicconsulting/168791/GallupPurdueIndex_Report_2014_050514_mh_LR.pdf