Upon "vote your conscience" becoming popular, you could see what it meant to those using it. The phrase means, "do not vote along party lines," or, "vote for someone outside of your party." By the way it is used, it has little to nothing to do with the conscience. Since the concept of the conscience is biblical, the person saying, vote your conscience, sounds like he's encouraging something biblical. Someone against "voting your conscience" also sounds like he's against doing something that must be biblical.
How would someone vote his conscience? The conscience doesn't provide information for voting. The conscience warns someone based on his highest perceived standard. The conscience doesn't tell someone how to vote.
I won't be relying on my conscience for my vote in November. I don't remember ever depending on my conscience in any election ever. To vote in a democratic republic, someone should study the candidates and the issues, developing a checklist for the pros and cons of each possibility. My priority for voting is what is best for the country.