Was there a sign for maximum capacity, and how did they know they reached it since I've never seen them count people who? Or did it simply look too crowded? I did a quick search on united.com & couldn't find any language about denying access, but I could have missed it. Local fire codes would take precedence, but I'd like to know how they determined it was full.

If you are still there, ask for the entry denial in writing, so you have some evidence that you were denied entry. I could see customer support saying, there we have no evidence you were denied entry.