On my cygwin environment the docker cert path is actually set as
DOCKER_CERT_PATH=C:\Users\user\.docker\machine\machines\default
and docker seems to work fine. I've just tried
export DOCKER_CERT_PATH=/cygdrive/c/Users/user/.docker/machine/machines/default
and it does indeed give me the same error
$ docker version
Could not read CA certificate "\\cygdrive\\c\\Users\\user\\.docker\\machine\\machines\\default\\ca.pem": open \cygdrive\c\Users\user\.docker\machine\machines\default\ca.pem: The system cannot find the path specified.
My docker version is 1.10.1, if the results are any different.