Using DAVDroid 0.5.12-alpha the connection to my DAV server is impossible.
What happens on the DAVDdroid side :
I/davdroid.QueryServerDialogFragment( 7393): onCreateLoader
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "CONNECT dav.anonym.org:443 HTTP/1.1[\r][\n]"
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "Host: dav.anonym.org[\r][\n]"
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "Proxy-Connection: Keep-Alive[\r][\n]"
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "[\r][\n]"
E/EnterpriseContainerManager( 691): ContainerPolicy Service is not yet ready!!!
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 << "HTTP/1.1 200 Connection established[\r][\n]"
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 << "[\r][\n]"
F/davdroid.SNISocketFactory( 7393): createLayeredSocket should never be called
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "OPTIONS /card.php/ HTTP/1.1[\r][\n]"
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "Host: dav.anonym.org[\r][\n]"
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "Connection: Keep-Alive[\r][\n]"
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "User-Agent: DAVdroid/0.5.12-alpha[\r][\n]"
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "Accept-Encoding: gzip,deflate[\r][\n]"
D/ch.boye.httpclientandroidlib.wire( 7393): http-outgoing-3 >> "[\r][\n]"
There are no logs on the server that the OPTIONS http request was ever made.
Using the DAV server is still ok from Gnome Evolution. I have also checked that no firewall rules was forbiding ma phone to connect.
It seems, but I am not sure, related to the upgrade of OpenSSL to 1.0.1e-2+deb7u6