On the one hand, of course, it's not up to Congress. One way or another: it depends on an interpretation of the 14th Amendment. So if it's to be changed, the change has to come from the Supreme Court. More generally, though, I don't think I've ever heard it seriously suggested that the 14th Amendment does not in fact guarantee citizenship to every one born in the USA.
One would think for these folks that the plain words of the Constitution would be enough to settle the matter. But apart from the Constitution and beyond all the ethical and prudential reasons why birthright citizenship is simply the right policy for the country to follow, consider what it protects against. In many European countries -- Germany being perhaps the best example -- you have substantial populations of stateless people. In Germany it's the Turks who came legally or illegally as guest workers, then had children. And these children were born and raised in Germany, speak German, but aren't German citizens. They aren't Turkish citizens either. And culturally they're no more Turks than Hispanic kids born in Los Angeles whose parents happened to have been born in Mexico. (ed.note: As TPM Reader JMN notes, the law was liberalized substantially in Germany about a decade ago, though it still does not allow citizenship for all born on German soil.)
It's a huge problem, not just morally but also in terms of nationality, stability and civic cohesion. It's both cause and effect of Europe's comparative inability to absorb and acculturate immigrants.
In all likelihood this is pure theater. Though nothing's for certain with the current Supreme Court, I doubt very much that any change is coming on birthright citizenship. It's black letter law in the constitution. But it's another sign of the darkening political horizon in the country, along with routine protests around mosques and all the rest.