What could cause chest tightness and squeezing after but not during exercise? Isn't angina type pain typically during increased exertion?

Yes but. You're correct, however there are exceptions to every rule. If you're having chest tightness and squeezing, it could be an atypical form of angina, but could also be a type of asthma or muscle spasms. It would be best to talk to your doctor about this.