Difficult question. I think that anyone who thinks that this one is really clear cut is missing the point - a pretty tough balancing of interests is involved.
My opinion, for what it's worth...
1. "It's Nigel's plane and you do what he says". BA's private property has essentially been opened up to the public for the purposes of commercial gain. This immediately places some limitation on what they can prohibit - the state (whichever it may be) has a greater regulatory interest than would be the case if I prohibited certain speech in my own home.
2. "It's like shouting "Fire!" in a crowded theater - you can't just say whatever you want and expect it to be protected as free speech". Absolutely correct - the problem is in determining what is legitimate and protected speech and what is tantamount to shouting "Fire!" in a crowded theater. The distinction which Justice Holmes had in mind when he wrote that passage, and indeed the distinction which he never quite got the U.S. Supreme Court to accept, was the distinction between words and action. When someone's words are just words, even if we find them irritating or offensive, we must tolerate them. When they actually cause adverse reactions with real risks, or incite certain undesirable behaviour, it may be legitimate to limit them.
If something is offensive enough or irritating enough, it may upset people so much that it has crossed that boundary, and we can perhaps legitimately limit it. It may even cause certain types of more tangible harm. Hence the legitimacy of prohibiting racist speech.
With this in mind...
3. Asking Gilmore to take off the pin was unnecessary - it had not been noticed and was not inciting any passengers to misbehave. It was not upsetting anyone or making them feel uncomfortable. A little hypersensitive and paranoid, I think. However, if the badge was having such an effect on pax, there might have been a case for making the demand, although only if it was causing *real* concern.
4. Once Gilmore had been asked to remove it, it had been brought to the attention of all and, more importantly, he started behaving like an arse. He could have removed it and complained (in my view legitimately) at a later time, but he chose to misbehave and cause a problem. If the Captain decided to remove him for this reason, well done. His political speech had turned to action which would have got him removed from the flight in any event (i.e. regardless of whether he was engaging in political speech). If he was removed because he refused to conform to the Captain's order to limit his speech, that would be illegitimate.
Of course, there is a fine line between getting thrown off for your behaviour and getting thrown off for expressing a view that leads to that behaviour - this is why it is such a difficult issue.
I admire the point Gilmore was trying to make, but the way he made it affected too many other people, and could have been made in a manner that was less selfish but equally effective. I think that he behaved, if the facts are reported correctly, like a complete arse!
Cheers
Flaps_45