Here is how Quinnipiac asked the question, first they set it up with this question:
They then followed up with this:
So again they didn't actually describe Obama's executive order and the previous question may actually lead many to think Obama's order was to let every illegal immigrant stay and become a citizen. Again this is why the Hart poll (which described how the order was limited) comes up with a different result.
In a way this is not unlike Obamacare. Polling shows that when asked about it in general, a plurality are opposed to it, but when you ask about the details, majorities like the features of health care reform. And most don't want to see it fully repealed, which is the parallel here. How many Americans want the government to spend the resorces required to deport 11M undocmented, including the 5m who parents of US citizens and have been here 5 years (aka the ones in Obama's order)?
That is the trick for the GOP in 2016. How can they find a way to be against this order and against immigration reform but not be painted as being for massive deportation.
That or being against the order whilst supporting legislative action.
This is the biggest thing people need to understand about the messaging impact. In 2006 and 2007 majorities and pluralties of about 10% to 15% supported a path to citizenship, but the bill to actually do so in 2007 was deeply unpopular. The biggest difference was in 2006 and 2007 there was a focused and energized effort, not unlike what happeend with Obamacare, to label and define it as Amnesty, on Fox, talk radio and even some of the hosts on CNN at the time like Dobbs. They also highlighted the increases in legal immigration and polling shows economically motivated opposition to that as well.
In 2013 you had Rubio running ads on talk radio describing the increased enforcement and downplaying the path, and also describing the path in the most innoucous terms possible. That is why the polling looked so rosy, largely because the messaging was so lopsided and the opposition was marginalized more effectively in 2013 then 2007 (As I said after 2012 the long knives were out and the COC was/is convinced that it is there path to a four year majority of pro-biz shill domination as well as the continued benefit of cheap labor).