I know exactly what you mean. Especially shopping, the stress, the people, the noise all get too much and i can't wait to get out.

I wouldn't say that i get headaches - but panic attacks? Yes, absolutely!

I get crowd panic too. Not as bad as it once was though. Long ago at a blues festival in Chicago that reportedly had A MILLION PEOPLE!- disgustingly hot day- I SNAPPED! I could no longer stand trying to swim through the crowd with people bumping and touching me. I bolted, pushing people out of my way like a football player. I kept running until I found an open space where I could breathe. I am normally very quiet, very peaceful, this was bizarre to witness myself doing this! Too many people in Chicago, glad I don’t live there anymore. Nothing like that ever happened before or since.
Sometimes I will stare at the ground right in front of my feet for a few minutes to quell the overwhelm. “Square breathing technique” helps me a lot.
Breathe in four a count of 4, hold for 4, out for 4, pause for 4, repeat as needed. ✌?

It doesnt present itself as dizziness or anything for me, but definitely a sense of pressure or urgency in a crowd. In a busy shop, as that understandably seems to be a common example, I'll get the sensation that the whole crowd are thinking ffs just choose your stuff, pay and get out of the way now. You described it as a squeeze, that seems to fit. I've never found it to be a major problem to be fair, but I know it can be crippling for some.

Yes, it's a form of a panick attack. For me, I get light-headed and dizzy if I'm feeling overwhelmed in a crowd. I have to find a place to sit down or at least something to lean against until it passes.

It usually comes from the noise of a crowd. It hurts my ears and my head