Governor Brownback has said he will defend our Kansas Marriage Amendment. This article says he is can do that.
For a governor to stand up and refuse to cower to a federal court would not be civil disobedience at all. It would be constitutional obedience-obedience to the Constitution and its provisions in the 9th and 10th amendments, obedience to his own state constitution, and obedience to the oath he took before Almighty God.
http://www.afr.net/articles/stand-against-judicial-tyranny/