The Answer is Simple: Make Me an Offer I Can't Refuse
Our generation may be self-centered, and may have questionable taste in music, but thanks to the economy we've all had to learn the value of a dollar. So if you want more millenials to sign up for health care plans, you're going to have to make a persuasive economic argument.
Maybe we're not well-informed enough. Show us how the insurance premiums compare to the healthcare costs that we risk without coverage. Some of us will sign up; others will decide that catastrophic insurance is enough, or choose to forgo coverage altogether.
But if that doesn't work, there is really only one other option - to recalibrate the penalties such that it will be cheaper for us to participate than to opt out. If the ACA is going to succeed, the penalties and premiums must reflect the law's actual costs.
Don't appeal to our patriotism, or ask our mothers to nag us about it. We will respond to the incentive structure that you design. If you want us to sign up, then make us an offer we can't refuse.