I'm no experience with Jenkins, I'm currently researching different options for PHP & JS automated unit testing with Jenkins.
I've come across Chutzpah (which uses PhantomJS's headless WebKit browser) but:
There's very little documentation on Chutzpah. Although it does state on the Chutzpah homepage that it can be integrated into the TeamCity continuous integration server.
It is possible to use Chutzpah with Jenkins and with the 2.1 release of Chutzpah it is easier. Chutzpah's command line client can now take /junit argument that lets you specify a file name to output a junit-xml compatible file to. You can use Jenkins to pick this file up and report the test results.