I just watched this little video from the Christian Medical Dental Society . We all have to get along in a diverse society, and forcing people to do things they don’t believe are right isn’t the way to go. Neither does conscientious objection always have to do with with religion. There have been longstanding feminist concerns about the birth control pill, for example.
Since freedom of conscience and religion is enshrined in our constitution, I’m not really clear on how this case can lose, but I’ll leave the prognosticating to the lawyers.