Your argument is with the Constitution. Coddington has no right to inquire about a public employees religion or political leanings and if you think he does you are on the wrong website.
He does have a right and a duty to ensure that his employees conduct science without an agenda. I don't trust the conversation as related in this piece (which clearly has an agenda), for it is clear that Coddington is violating employee rights. However, if Coddington had asked: "Is he trying to insert either his religious or political beliefs into his work?" Coddington would be doing his proscribed duty.