There’s an interesting Slate article discussing the question. An excerpt:
Rep. Dave Camp, the Republican chairman of the House Ways and Means Committee, is Roman Catholic, and he represents a district, the Michigan 4th, with few Jews. But as anyone with access to Twitter, Facebook, or the rest of the Internet can learn, Rep. Camp has a big Jewish problem. And it’s one he may be powerless to solve.
The congressman is under attack because of his aide, Aharon Friedman, an Orthodox Jewish graduate of Harvard Law School. Friedman has been legally divorced from another Orthodox Jew, Tamar Epstein, since 2010 — but has refused to give his ex-wife a get. In Orthodox Judaism, this is the document that a man must give to his wife in order for a religious divorce to go into effect. So long as Friedman refuses to give a get, Epstein cannot remarry within the faith and is considered an agunah, or chained wife.
Epstein’s limbo status has sparked an outcry in the Orthodox world…. Insisting that Friedman’s conduct amounts to domestic abuse, [Epstein’s supporters] have used the Internet, including social media and the petition site change.org, and the national media to demand that Rep. Camp pressure Friedman to religiously divorce Epstein….
Note that Congressmen, as employers, are bound by the constitutional constraints that apply to the government generally. See, e.g., Davis v. Passman (1979). And while high-ranking Congressional employees are excluded from the normal constitutional protections against government discrimination based on employees’ political activity, I don’t know of any authority for the proposition that such employees are excluded from the normal protections against government compulsion of religious conduct.