I succeeded in posting a message with an attachment to Quitter.no.
This required two changes:
1. Set apache HttpPost request to HTTP 1.0 protocol:

HttpPost request;
...
request.setProtocolVersion(HttpVersion.HTTP_1_0);
When creating Multipart post request I'm providing as an input for an attachment not a InputStream (as for HTTP 1.1, which causes chunked encoding), but an array of bytes, which I have to create from the same stream beforehand. This is why content length is known.
BTW, I still don't set content length header anywhere in AndStatus code. This is done at HTTP library level.

The only thing left is to add the "USE legacy HTTP protocol" option to a "Social network".

@moshpirit@quitter.es @andstatus@loadaverage.org @hannes2peer@quitter.se @mmn@social.umeahackerspace.se @lnxw48@fresh.federati.net @knuthollund@quitter.no