The comments have already pointed out that the President of the United States is still a citizen, and all of the rights of a citizen are still protected for them. Additionally, the Administration is allowed to take policy positions which are antagonistic to a person or group's cause, even if that group is practicing their rights to express their views legally. To give a different example, the President and his administration may denounce the position of a group of Neo-Nazis marching legally. So, any argument that the President is acting in an official capacity while making antagonistic comments also probably fails, as the Administration is allowed to take a position on any issue they deem worth taking a stand on.
As noted in another answer and in comments, the applicable laws appear to be 18 U.S. Code § 227, which provides for punishment of government officials who attempt to influence employment decisions through official acts for political purposes, and 42 U.S. Code § 1983, which provides for civil action when a person deprives, or causes the deprivation of, another person's rights under color of law.
18 U.S. Code § 227 likely does not apply for two reasons
- The President may show that his conduct was not purely for political purpose
- The official statements made do not qualify as official acts per McDonnell v. United States, as they are not
a decision or action on a "question, matter, cause, suit, proceeding or controversy"; that question or matter must involve a formal exercise of governmental power, and must also be something specific and focused that is "pending" or "may by law be brought" before a public official
42 U.S. Code § 1983 might apply if the official statements were found to be acting under color of law, but I think the statements made so far will fail to meet the qualifications for this statute. Blair v. Bethel School District gives three qualifications for conduct that would allow recovery under this statute:
(1) he engaged in constitutionally protected activity;
(2) as a result, he was subjected to adverse action by the defendant that would chill a person of ordinary firmness from continuing to engage in the protected activity; and
(3) there was a substantial causal relationship between the constitutionally protected activity and the adverse action.
The President's and Administration's official speech appears to fail the second criteria. First, it is questionable whether it qualifies as an "Adverse Action" - in Blair, as well as Hartman v. Moore and Gibson v. United States, the adverse action against the Plaintiff caused actual damage or indignity. However, even if we assume the official speech qualifies as an adverse action for the purposes of the statute, it still appears to be permissible for effectively the same reasons as the first and third arguments presented in the decision:
First, the adverse action Blair complains of was a rather minor indignity, and de minimis deprivations of benefits and privileges on account of one's speech do not give rise to a First Amendment claim.
The actual effect of the Administration's speech has a minimal direct effect on the players it speaks against. The decision further states:
The most familiar adverse actions are “exercise[s] of governmental power” that are “regulatory, proscriptive, or compulsory in nature” and have the effect of punishing someone for his or her speech
Official speech by the Administration is not "regulatory, proscriptive, or compulsory in nature." While this is not a complete definition for "adverse action," it gives a sense of severity, which official speech does not appear to meet.
Additionally, the President's right to speech, and the Administration's authority and need to make official speech as directed by the President is a competing interest in this case, as was the interest of the Board in Blair:
Third, it is significant that Blair isn't the only party in this case whose interests implicate First Amendment concerns. To the contrary, we assume all of the Board members have a protected interest in speaking out and voting their conscience...
The decision does note that:
The point isn't that the vote against Blair was protected speech simply because it was expressive. Almost all retaliatory actions can be said to be expressive, including those that are manifestly unconstitutional. But, while Blair certainly had a First Amendment right to criticize Seigel and vote against his retention as superintendent, his fellow Board members had the corresponding right to replace Blair with someone who, in their view, represented the majority view of the Board.
Similarly, it's probable that a court would find that the President's right to speech and their Administration's corresponding authority to speech against the players' right to protest is equally weighted or even weightier, such that stifling the official speech is as bad or worse than the alleged chilling effect of the speech.
Some examples of things that clearly would fall afoul of 42 U.S. Code § 1983 would be the President or the Administration misappropriating funds to use to pay NFL teams not to hire players who kneel during the national anthem, or signing an Executive Order preventing players who kneel during the national anthem from playing - in both cases, they are taking actions which fall outside the powers of their office, which would qualify as acts made under color of law and clearly chill the players' First Amendment rights.