Has anyone warned you about age 3 yet? It was much worse for us than 2 with our son. Our daughter turned 2 a few weeks ago, and she's throwing more tantrums lately - especially with sharing toys (even when she's not playing with them).
One of the things that worked with our son was to ask him to "show us" when he needed something if we couldn't understand.
Our pediatrician recommended diversion as his preferred tactic in those situations. Doesn't work for our daughter. But, telling her to stop crying has been strangely effective. She was doing it last night as we were all playing outside. Another kid took a ball out of a bucket she was playing with, and she melted down.
I grabbed her wrist, sat her in the grass and sternly told her to stop crying, start sharing or she would go inside and not play with the other kids. She whimpered, and it was over.
I had a similar situation to yours with our son at 2.5 in Target. He wanted a certain Matchbox car, and it was right after Christmas, so I told him no. He melted down repeating "I want red bus" over and over and over. I walked all over the store with him screaming, and everytime he said it, I responded with, "I know you want the red bus and so does everyone else here". I know I got the evil eye from everyone, and I did the best I could by simply ignoring him and not giving in.
Good luck. He's exerting his independence and learning your boundaries. I applaud you for standing your ground.