Interesting question, but this is pretty broad. I haven't seen an unit testing tool yet which does all the backend, midend and frontend testing at once. I checked this list, but no one does it all. There is however one generic approach, xUnit (fully automated testing).
The major blockers are that backend testing is platform/DB specific and that midend testing is programming-language specific. And then yet to combine those limitations with frontend testing. The tool should then have to support almost all languages and platforms the world is aware of. I don't think such tool would ever be available in open source world. It's an utopia.
In theory it's indeed possible to have a testing framework with plugin capabilities, maybe based on the xUnit ideology, but one has yet to write/invent/opensource it. You? It's indeed a hole in the market.
At any way, for frontend unit testing (HTML/CSS/JS/forms) I would recommend Selenium, or if you have the money, TestComplete. I have seen it been used at IBM several years back, it was awesome to see it in action and the testers were very happy with this.
For midend unit testing (programming code, the business logic), just continue with the programming language specific unit testing tools like PHPUnit for PHP and JUnit for Java.
Regarding backend unit testing (DB, filesystem), I've used PGTap for PostgreSQL, but there are also generic DB tools available for this such as SQLUnit (which is however last updated almost 4 years back...). For the local disk file system conditions you'll have to grab platform specific scripting languages.
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