Am I the only one that clicks "remind me later" 99% of the time my computer asks me if I want to do something?
Behavioral economics, from what I can gander, has a few central tenets. People prefer to choose the "middle" option, so anchor them to a really high price (like a $99 "Titanic" sushi dinner) and they'll pay more. People tend to neither opt in nor out of programs, so if you want organ donors, make them opt out. And finally, people are rampant procrastinators!
That third point is why the "remind me later" button is so poorly designed. It allows you to put off the problem until later at low cost. If the software engineers want you to make the updates then they should say: "I don't have time to update now." And if they don't care if you make the changes, then why would they waste your time asking?
My suggestion isn't great, but at least users may consciously think about how much time they have. And if you combine this with a button telling them that updating will take less than 10 seconds, that would be a powerful 1-2 punch.