I don't know if Cruz is responsible for stopping it, but Cruz did clearly state in 2014 that the House Plan was a legalization amnesty, and that he was against it.
After the 2014 midterm elections, two things probably killed the chances of an amnesty passing. First, the defeat of Eric Cantor put a chill on many Repubican amnesty pushers in the House. And second, Obama's illegal executive amnesties killed whatever remaining chance amnesty bills might have had during 2015 and 2016. And, of course, that's still going through the courts.
But Boner never gave up. He kept trying to revive it during 2013 and 2014. Here is an interesting article where Boner was pushing amnesty in Ireland during July of this year, and the article summarizes the state of amnesty during this year pretty well.