Probably just continuing to talk to myself here, but I'm going to backtrack on that remark about iOS 6.1. I just upgraded to a new Mac Pro running 10.8.2 and experienced the same problem with my website. But another webcam site was working. So I fiddled around with my (php) mjpg code and finally it started working. And then I tested it on my iPhone and it worked there too! But that third-party webcam still doesn't work on my iPhone. Go figure. But it doesn't bother me now that my site is up again.
What worked in the end was to enure that the actual jpeg images are surrounded by two cr/lf pairs. The output is sort of like this:
HTTP/1.1 200 OK
Date: Wed, 06 Feb 2013 18:32:11 GMT
Server: Apache/2.2.22 (Unix) DAV/2 PHP/5.3.15 mod_ssl/2.2.22 OpenSSL/0.9.8r
X-Powered-By: PHP/5.3.15
Transfer-Encoding: chunked
Content-Type: multipart/x-mixed-replace; boundary=--myboundary
--myboundary
Content-Type: image/jpeg
jpeg-frame-here
--myboundary
Content-Type: image/jpeg
jpeg-frame-here
--myboundary
Content-Type: image/jpeg
...