I'm the head of software engineering at a small company and was a technical director at an MNC previously. I've hired hundreds of programmers.
I regularly hire EEs as programmers, but not for web development. Web development is mostly the bastion of very nimble, hacky types. As others have said, it's frequently more about putting together a reasonably elegant hack in a short period of time.
I hire EEs for board support and other embedded development. Those are the places where real engineering skills are the most useful. I don't want my BSP full of dirty hacks or hard to find/duplicate bugs. I want code that is planned, organized and well executed. That's exactly (in my experience), what I get from engineer coders.
The exception to my above generalization about web development is Java. Java backed websites (JSP and the like) are mostly developed by engineers and are used by large companies. If you want to maintain your engineering mindset and build websites, Java dev as a nameless drone at a big company is the way to do it.