I agree with Joe that this is something you probably can, but shouldn't, refuse. And yeah, it's one of those things that I as an introvert would also rather avoid. But it's probably better not to dodge it.
It's a bit like fraternity hazing. Not quite as juvenile, but kinda reminds you. You don't get to be the cool kid by trying to weasel out of it. It can be a bit embarrassing, but everyone else has also been through it and it's okay to be a bit embarrassed.
I do think that if this sort of thing is required of newcomers who aren't that high on the totem pole yet, that it would be a good gesture if someone established at the company also has to do such a dance act, to sort of share the awkwardness. (And I'm not talking about the manager who happens to be a seriously good dancer, showing off. I'm talking someone regular who shows that they're human too, and that it's all in good fun, and they're not afraid to look funny at a party.)
Also, if you decide to do this, do make sure you practice. Ask if there's someone at the company who's good at this who's willing to help you practice. That person will also be able to help you mentally prepare for it. The point of practice isn't to suddenly become a star dancer, but to feel mentally ready for it.
It's probably going to be awkward. But not maliciously awkward. Everyone else who's had to go through it will laugh a bit but ultimately sympathize with you. Try to enjoy it, even if it's not something you'd volunteer for normally. It'll help you become part of the gang.
EDIT
This answer is getting a lot of downvotes, but almost as many upvotes. I guess I struck a nerve, and that the answer needs improvement.
Let me start by saying that I don't think this office policy is a good policy. I would hate to have to do such a dance act myself. I'm also an introvert, in a world that sometimes seems built for the pleasure of extroverts. And I've always been nervous around these "put the newcomer on stage" things.
But I also try to be pragmatic. I could take a principled stand and refuse to perform, but I'd probably get a reputation as a sourpuss. Maybe it'd cause some change in the company, but the chance of that doesn't seem very large, because as a newcomer I don't have a lot of sway yet. I think that if I tried to fight this head on, it'd just give my career in the company a bad start.
The OP reports that seniors have also asked HR to not have to perform but that HR required it. I suspect that HR has an attitude of "oh, it's just a dance, it's fun, don't be a childish whiner". As a newcomer, you don't have the social power to challenge this. But the OP's colleagues who are by now seniors have more power now than they had when they joined and they had to dance. If you want to effectively challenge this policy, you need help from people with more status in the company.
That's why I said that what should happen is that the OP's manager (or a senior colleague) gets involved. It's easy to point and laugh at the insecure newcomer. Not so easy and risk-free to do that if the newcomer is dancing together with someone with some status in the company. A good manager would recognize this situation and use their social power to protect their junior. In this case, by going on stage and shaming HR.