This seems to be the same problem as https://forums.bitfire.at/topic/1100/v1-0-9-http-1-1-415-unsupported-media-type
Please use DAVdroid 0.9.0.1.
My server delivers the right cert when checking with chrome or the ssl-check-service you recommend. It also worked in previous versions of davdroid. With the new http lib it uses the default host cert and fails.
Duplicate of #205
I am runnung 4.4.2 with the moat recent davdroid version from googles appstore.
There is already a bugfix release 0.5.10.1 that fixes the sni issue. Wait some hours and you will receive the update automatically.