The constitution guarantees certain rights. What that means is that government can't deny those rights.
The important point is that "the government" is not some disembodied thing that is ruled by the constitution. The government consists of people. It consists of civil servants and elected representatives going about their day to day duties. So when we say that the government can't deny certain rights, we mean that these individuals, while acting in their governmental roles, cannot deny those rights.
If we allow these individuals to pick and choose which rights they will respect, then we might as well just throw out the constitution because we've then made our guaranteed rights conditional on who you encounter in the bureaucracy.
When someone runs for office, or takes a government job, they need to realize this. They need to realize that they are bound by the constitution as well as by laws enacted by congress. They further need to realize that those laws could change at any time, sometimes through legislation, and sometimes through court cases, and that they will then either have to obey those laws or resign from their positions.
There is nothing brave or noble about what Kim Davis et al. are doing. What they're doing is attempting to set themselves up as an arbiter of SCOTUS' decisions, picking and choosing based on their personal beliefs. Since she sincerely feels that she can no longer carry out her required duties, she should resign and give up her $80k salary.
That would be brave (but still misguided, obviously).