A group of House members [1] have introduce the "Legal Workforce Act" ("LFA"), which would require the use of eVerify by most workers nationwide. That would presumably make it difficult for companies to employ illegal aliens. However, because it contains some concessions to employers and was written with the input of the US Chamber of Commerce (which supports illegal immigration, see the link), some are claiming it's a hobbled bill that wouldn't be enforced. That point of view is given more weight because it forbids some state immigration-related enforcement.

Those on the pro-LFA side include FAIR (link) and Mark Krikorian of the Center for Immigration Studies (link).

Those on the anti-LFA side include Kris Kobach (link) and Russell Pearce (link).

You can read the PDF with the bill here and a PDF with a summary here.

I'm tilting slightly to the FAIR side of things, but it's important to note that bills like this might not work or would be less effective because few are willing to do the necessary, fundamental work first.

Namely, discrediting those who enable, support, or promote illegal immigration. Even if the bill passes, the groups and people at those links will still be around doing what they've been doing largely unimpeded for years. The bill being passed isn't going to prevent those like George Soros or the Koch family from giving millions to groups like the American Civil Liberties Union, and it's not going to stop the ACLU from fighting against most forms of immigration enforcement and frequently winning.

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[1] Lamar Smith, Ken Calvert, Elton Gallegly, Jason Chaffetz, Brian Bilbray, Ed Royce, Gary Miller, Sue Myrick