Would an official who directly curtailed a Right, as enumerated in the Constitution, be considered a High Crime or Misdemeanor?

If the question is asked as a justification for impeachment. "High crimes and misdemeanors" are whatever the House of Representatives decides "High Crimes and Misdemeanors" are when they draft articles of impeachment. Impeachment is not a criminal proceeding. It is a political act. The only questions are will the House reach the number of "aye" votes to refer the charges to the Senate? and will the Senate reach the number of "aye" votes to convict and "punish" the subject of the proceeding with any or all of the options they have.

Back to your question:

1) No single person may enact a law.

2) The Constitution makes no allowance for establishment of a state religion. It expressly forbids such an act.

3) If such a person acting as head of the executive branch or a regulatory agency tried to claim they were enacting such a regulation under existing law, it would go absolutely nowhere under the federal rule making process that is required of regulatory bodies in the U.S. unless Congress approves. [Return to step 2)]

Finally, if all else fails

"We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.--That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, --That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed. But when a long train of abuses and usurpations, pursuing invariably the same Object evinces a design to reduce them under absolute Despotism, it is their right, it is their duty, to throw off such Government, and to provide new Guards for their future security." - Declaration of Independence, Thomas Jefferson.

No less true today than 246 years ago.

Tim,
People are subjected to it daily--workplaces limit discussion of certain topics; to get into a military you must subject your vehicle to search at any time for any reason [remember gate checks for "drugs"].

Okay. The example is a bit facetious. With that said, the President could sign an Executive Order, create an Agency (or an office withing an Agency, or direct that a particular Agency create an office within it), with a net effect of curtailing a Constitutional Right. That might need some cooperation from Congress, but let's say the President has an amenable Congress.

Admittedly, impeachment is a political, vice legal, action. Nonetheless, if another Congress (different from the one that cooperated with the President, earlier) wanted to construe such actions as impeachable, then would it hold much water in the Public's eye?