As an older worker, I'd say the mistrust issues of younger workers are entirely justified. If that is a problem, corporate management is hoist on its own petard.
If you wanted Gen Y to trust you, you should have treated Gen X better--or at least hidden the mistreatment better.
As for the entitlement, that's probably on their Boomer parents/grandparents. You could shove them down and kick them in the teeth until they become bitter and jaded, just like Gen X. But then you're back to the trust thing again.
So if I may humbly suggest, start treating your older workers better. First, have some to treat better. Then, give them back all the crap that Boomers once enjoyed, but which later got stripped from standard employment arrangements, like actual vacation weeks and sick days, longer-term employment contracts, career ladders without important rungs missing, viable retirement plans, job security, etc.
The younger workers were born into the networked world. They rely on friends and family to judge your company's reputation. And the source of that reputation is largely going to be people that worked for you in the past. Those older people have all been screwed by their employers in one way or another. Kids are usually naive, ignorant, and inexperienced, but they aren't stupid. They see what happens to uncles and cousins and friends of their parents.
Sadly, there's not much an individual manager can do about any of it. Those young workers are coming in expecting to get stabbed in the back, and one day, your CEO or Head of HR will call a meeting and issue you your knife--or maybe your axe. Those kids know it's coming; they have seen it happen over and over again to other people. So maybe validate the mistrust, and admit that the company doesn't care about them. Let them know that the one and only way to keep a job is to be productive enough to pay their own wage, and then some, and that everyone should plan to eventually jump ship near a port, before the captain gets the chance to throw them overboard in the middle of a crossing. Recognize that a mercenary attitude puts self first. That makes the manager's job harder, of course. Why should a mercenary work any harder than necessary to earn the paycheck?
I would (as a non-manager) use discretionary cash bonuses to reward good employee behavior, with intermittent reinforcement. Cultivate trust in you, personally, rather than trust in the company. Punishments and the threat of termination simply won't work. Just put everyone on a secret carrot schedule, and when it comes time for the next reward, set up a plausible reason to hand it over.