What you should ask yourself in this case is not whether you think people should have access to birth control, but whether you should be required to do things that violate your deepest convictions. Should a soldier be required to torture prisoners, for example? Should he refuse to do so if ordered? Should a liberal corporate peon be required to contribute to the Republican Party?
On further reflection, I find such arguments unpersuasive because the assume that dispensing prescription drugs is just another job. It is a tightly regulated profession with stringent entry requirements. Also, it is silly to compare it to being required to torture, which is both illegal and universially considered reprehensible.
When I see a blogger argue that being able to refuse to dispense drugs is a matter of "freedom" I have my doubts. John Brown argues that "a private business should have the right to decide what it wishes to sell and what it doesn't." I agree, but if he thinks they are simply private businesses, he should open a pharmacy and see if any other pharmacies have moral qualms about shutting him down. He also ignores the probability that "conscience clauses" will be used by druggists against their private employers who wish them to fill all prescriptions.