Having a conscience, by itself, means nothing in particular. A conscience must be grounded upon principles, and it must be well formed.
Many people have interpreted the Church's teaching that one should follow his conscience to mean that if an individual doesn't agree with Church teaching, he can legitimately ignore what the Church teaches and follow their own individual conscience. This kind of individualism resonates with our modern age, especially in America.
We live in an age when personal freedom is exalted beyond all other rights. Catholic politicians frequently promise their constituents that they will not take their orders from Rome, that they will follow their consciences. But what defines the conscience of a person who professes a faith and rejects its most fundamental tents?
Could a politician profess the Catholic faith and then advocate the murder of innocent human beings? If one were free to follow one's own individual conscience on any topic, then we would have to say yes, one might advocate murder of innocent human beings as long as it doesn't violate one's own conscience. Clearly, there needs to be some guidance to our consciences. The Church provides this guidance persistently and clearly through its teachings. Therefore a faithful Catholic politician who advocates the trampling of those rights which our faith teaches us are inviolable, is either violating his conscience, or he does not have a Catholic conscience.
No leader worth his title ever personally opposed a great evil while advocating it.