There are limits to the constitutional guarantee of free speech. Child porn is one of those limits; saying “fuck cheer” is not.
Without know the justification for the rule, yes. But I suspect there’s actually a pretty good justification for why schools should be able to infringe on a students freedom of association in this way.
The justification offered here has been, “She’s not be resepectful enough.” “Not being respectful,” is not a sufficient reason for a government proxy to limit someone’s constitutional rights.
Assumes facts not in evidence.
There are coaches who will say, “By publicly questioning my decision, you’re undermining my authority with the cheer team. And we don’t have an ATMB. You’re suspended.”
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.That bit about the ATMB is a joke, just in case a mod doesn’t get it.
This pretty much sums up my opinion on the matter.
I would add one thought to the matter. No rights are absolute and your rights end where others’ rights and legitimate interests begin. In Canada, the Charter of Rights and Freedoms - a constitutional document - says that the rights contained in it are subject to such restrictions as can be “demonstrably justified in a free and democratic society.” What is and isn’t demonstrably justified is of course a matter of discussion. However, I think that minors / schoolchildren should not be held to higher standards (I.E. to greater restrictions) on freedom of speech than adults. And I fear that in practice, they often are. This is also my issue with Tinker v. Des Moines. Do anti-war armbands really substantially disturb the course of a school day or was it more of an ideological restricition?

Assumes facts not in evidence.
The objection is sustained, but I’ve never known a coach not to kick your ass harder and respect your after questioning him…Just, IMHO.

No rights are absolute and your rights end where others’ rights and legitimate interests begin.
This, respectfully, is a giant pile of dog shit. Of course no rights are absolute. Nobody disagrees with that statement. This pile of shit is only used when someone wants to infringe on a core right contained in the text of the Constitution.

This pile of shit is only used when someone wants to infringe on a core right contained in the text of the Constitution.
… which is the position you’re arguing in this thread.

are you saying that the cheer squad members are members of an organization?
I would say yes, specifically:

members
of

the cheer squad
She was caught badmouthing the cheer squad to other members of the cheer squad. It’s frankly directly undermining the ability of the adult leaders to manage the squad.
So the cheerleaders can never question or disagree with the coach, then?
You keep question begging. Whether it is a First Amendment right is the argument.
I agree with you! I think using her membership in the cheer squad is a stronger argument than using her “membership” in the school as a student.

I don’t care what she said. If it’s off campus or not at an official school event then the school has no role or responsibility. It’s strictly a parenting issue outside of the few instances where speech is legitimately threatening enough to involve law enforcement.
The school does have a role and responsibility to make sure her participation in school-sponsored extracurricular activities is in the best interests of all participants; there is no “right” to be a cheerleader. If her behavior outside causes concern about what her behavior on the team might be, does the school still have to allow her the opportunity to screw up on the team, given e.g. that a failure to respect coach’s instructions could result in injury to her or her teammates?

So, if the principal arbitrarily decided to ban a student from club sports because he or she didn’t like their speech, that would be a problem, yes.
That’s not the question, though. If the student never even asked for permission, e.g., and still got booted from interscholastic competition, that’s the school imposing consequences for behavior that occurs outside of school hours and off school property.

Without know the justification for the rule, yes. But I suspect there’s actually a pretty good justification for why schools should be able to infringe on a students freedom of association in this way.
The justification offered here has been, “She’s not be resepectful enough.” “Not being respectful,” is not a sufficient reason for a government proxy to limit someone’s constitutional rights.
Look at page 12 of the Colorado PDF, the one with the form Durango Public Schools requires club athletes to sign. If you want to play for Durango, you have to make a commitment to attend a minimum number of practices and contests, to travel with the team, and to comply with the high school’s code of conduct. In other words, you have to make a commitment to be a responsible member of the team, not just an occasional dilettante. In other words, you have to respect the team.
That high school code of conduct is another interesting point. (Here’s a recent PDF version, although it may not be the most up-to-date.) It covers topics such as substance abuse, in or out of uniform and on or off school grounds. A violation can get you thrown off the team, even if it isn’t necessarily a matter for law enforcement.

So the cheerleaders can never question or disagree with the coach, then?
Question or disagree, sure? “Fuck cheer” isn’t a question, though.

Question or disagree, sure? “Fuck cheer” isn’t a question, though.
It’s a disagreement.

It’s a disagreement.
With what, though? The only possible thing I’m getting is that she disagrees with the coach’s ability to pick who is on the varsity team and who is on JV, and that’s reading a lot into two words, one of them an obscenity.
No, “Is this covered by the first amendment?” is the debate. The argument is where you explain why you think it does or does not. Asking you to supply an argument is not “begging the question.” Nor, for that matter, is pointing out that railing against a “pile of shit” that’s “only used when someone wants to infringe on a core right contained in the text of the Constitution” takes a remarkable amount of chutzpah when you’re the one in this thread taking the position that a core right contained in the Constitution should be infringed.

think that’s my point. The school wouldn’t have suspended her if she just said “Coach, your decision was bad. I should have made varsity!” The vulgar and immature WAY she disapproved of the school was the cause of it.
So speech is allowed if it isn’t vulgar?
“Fuck” is not ok.
“Bad” is ok.
What about “bullshit”, “crock”, “poopsicle”, “upsetting”, “disappointing”, “fragged up”?
“Poop on cheer, and Coach is a poopy head”?
“I am angry at cheer, and Coach has upset me”?
What if it had been expressed in emoji?
I want a bright line.
Not at all. It’s like saying I have a nuclear device in my basement and you have to prove to me that I can’t have it under my Second Amendment rights. That’s not how things work. You have to show that this girl has the right to flip off her coach and say fuck before she gets coverage under the Free Speech Clause.
I’m losing track of what argument you’re making here, UltraVires. I don’t think (could be wrong) anyone here has argued that the school didn’t have the right to throw her off the cheerleading squad (though obviously at least some Federal courts disagree with that for some reason). Some have stated the belief that the school was incredibly vindictive and petty to do so. There is debate about whether it would be permissible for the school to suspend her or inflict some other punishment that would affect her status as a student exercising her right to obtain an education, as opposed to her status as a participant in a voluntary extracurricular activity.
I don’t even see a student being kicked off the cheer squad, here. I see a student voluntarily quitting the squad, the coaches recognizing that, and then the student regretting it. At which point the coaches have no more obligation to let her back on than they do any other student.
“Fuck cheer” sure looks to me like “I don’t want to have anything to do with the cheer squad”.
I thought most of this thread was about whether or not she could be thrown off the cheerleading squad.