The US developed from an earlier kingdom, and the First Amendment enshrines the main issue that led to our departure from that kingdom. The underlying political premise has been that disagreement is to be dealt with rationally and not through force, such as where opinions contrary to those articulated by the government are squashed (in order to eliminate divisions).
There have been numerous laws passed in the US to outlaw "contrary" speech including "disrespectful" speech, and they are constantly being overturned by the Supreme Court. Texas v. Johnson, 491 U.S. 397 and U.S. v. Eichman, 496 U.S. 310 are two recent reaffirmations that such laws are unconstitutional. The only way such a law can work is if the Constitution is amended to in some way re-write the First Amendment, like this.
Outlawing indirect insults towards political figures ("not my Speaker of the House", "not my FBI director") would require an even more extensive suspension of the First Amendment. It is possible that at some time, a bill was introduced to outlaw saying disparaging things about POTUS, but I would be surprised if it got out of committee, because it would fail challenge in court.
The most-likely retrenchment on our freedom of expression is likely to be a flag-burning law, which has relatively wide support in the US. There are a number of interpretive problems associated with the key concept "physical desecration". Even more interpretive problems would arise if Congress were given the power (via an anti-disparagement amendment) to outlaw "disparagement of public officials". Can one simultaneously "respect the Office of the President" and "disrespect the holder of the office"?
As for specifically restricting the military, that's a challenging issue. It is a court-martial offense for a commissioned military officer to use contemptuous words against the President and Congress (10 USC 888), and by directive from the Department of Defense this also applies to enlisted personnel. SCOTUS in Parker v. Levy, 417 U.S. 733 articulated the Military Necessity doctrine, that "The fundamental necessity for obedience, and the consequent necessity for imposition of discipline, may render permissible within the military that which would be constitutionally impermissible outside it". So "not my President" probably is illegal for soldiers. This article reviews various First Amendment issues as they pertain to the military.