That does not mean that INDIVIDUALS cannot have standards that they want to uphold in their candidates - just that there cannot be an OFFICIAL qualification one must pass in order to become a candidate.
Progressives and atheists are trying to make a "religious test" that no religious person can be qualified for a public position (see Shawn Mitchell' 11/26/12 column - http://townhall.com/columnists/shawnmitchell/2012/11/26/lefts_newest_inquisition_deny_your_god_or_be_deemed_unfit). The furor over Richard Mourdock's religiously orthodox statements about rape and God's will is a case-in-point. The Left will only accept religious values that support liberal positions. All others are "crazy","fanatic" and "extreme".