Is this something I can go to HR about?

Yes, that's something you should go to HR about. You have the perfect vehicle for doing that since your boss wrote you up. You aren't being a whistleblower; you're appealing the write-up.

Don't be surprised if you win your appeal on the dress code violation, because it sounds like you weren't properly advised ahead of time, but lose your appeal on the insubordination. From what you've said, it does sound like you were insubordinate.

You said yourself you were annoyed, and so rather than take the direction from your boss, you argued you with your boss. Arguing with your boss over directions they're giving you is always insubordinate. It doesn't matter if your boss is right or wrong. It's insubordinate to stand there and argue with your boss instead of taking the direction and either appealing it properly with HR or saying you now stand advised that you must follow the dress code even when coming in for meetings and won't come dressed like that again.

If I were you, I might go speak to my boss next time I'm at work after both of our tempers have cooled and tell her that you've been thinking about going to HR because you don't want those write-ups in your file but then thought you might be able to settle it just between you two, saying that you apologize for arguing with her and tell her that in the future you'll be sure to follow the dress code for meetings. Your boss doesn't want to deal with HR. If you go in and offer her what HR would certainly uphold anyway, I think chances are good she'll rip up both write-ups.

A big part of the issue here is "I know for a fact I wasn't wrong."

You do have a reasonable complaint, but you've already taken it too far by refusing to sign the write up. HR will listen, but in the end you will not be satisfied. When you work for someone else, they're not expecting that kind of push back.

Sure you can go to HR and perhaps be a smart mouth with them. Clothing no problem winning but then you mouthed off.

Better start looking for another job because this one is almost at an end if you go over the bosses head.

And forget the argument that you were fired because of this incident because as an employer I KNOW they can find another reason for letting you go without even needing a reason barring a contract that says otherwise.

To accomplish your goal, you must be level headed and not emotional when you go to HR to report the two unjust write-ups against you.

I would simply state that you were away from home on your day off, taking care of your personal business, when you got the 60 minute notification of an unscheduled meeting. You had no choice but to attend as you were because otherwise you would not make it to the meeting if you had to go home and change your clothing. State that under the circumstances, you decided to attend the meeting rather than miss it.

If you are being paid to attend the meeting, then you are on the clock and that could make your argument invalid as if there is a dress code, you violated it. If there is no dress code and you were off the clock, I can see no real intent that you were being disrespectful, clothes-wise.

you have a reasonable case for HR

Go to HR, as long as you are in school you may get away with it. But when you get a real job don't even think about doing something like that. Oh by the way you were totally wrong.

Let HR settle it. I suppose that's really all you can do. I think in this situation you're not in the wrong. You'd be crazy not to at least bring it to their attention.

I think you have a case, so yes, you can speak to HR if you wish.

However, I wonder whether first trying to meet with your boss when you've both cooled down a bit might be helpful. Perhaps you can ask her, politely, why she was so concerned about your clothing and the fact that you pointed out you didn't have to adhere to a dress code off the clock. Be bewildered, express surprise and concern, say that the whole thing seemed unusual and mystifying. You didn't understand.

If she reacts badly to that, then perhaps you need to go to HR, but give her chance to rethink her actions first.

Yes, all so ask your HR about meetings not getting paid for. Explain to your HR about short notice.

Yes, you do have a reasonable argument for HR. If THEY don't do anything, contact an org like the ACLU.

Are you looking to be fired? He was right and you are wrong. It makes no difference that it "was" your day off. You were called in. You should NOT have showed up in sweat pants. It was a sign of gross disrespect - insubordination. Change your attitude, or expect more trouble.

Does it really matter? No matter who was wrong, do you really want to keep working for that boss? If you were "wrong" for being dressed wrong, you probably aren't the right person for that job. And if your boss was "wrong" for being upset with you, your boss is probably not the right person for you to work for.

Go to HR. It sounds like your boss wants to get rid of you and he will probably find a way to do it. Get ready.

You have my permission to go to HR.

Absolutely, go to HR and complain!

HR is a great place tot start. Union too if you have one.

Yes. If there is no dress code, she had no right to write you up.
You have the right to refuse to sign something that you do not feel is correct.