Intersting 14th Amendment argument. It ultimately fails because a stay at home order does not deprive any person of life, liberty, or due process of law. Under the Tenth Amendment, the state has the police power, and NC’s Emergency Management Act outlines the Governor’s powers. https://twitter.com/mikesadams/status/1254044400386220032

It's very complex and determines how the statutes are worded. It is feasible that there is a constitutional challenge to be made to some stay-at-home orders. But a stay-at-home orders are not inherently unconstitutional.