There is more to it than is open at the moment. I have yet to piece it together, fully, but it has something to do with his family, his role in Vietnam, and his role as a senator afterward.
It's "beyond treason" - Q is basically indicating that he has been stripped of all rights to his name, not just family name, but the given name as well. The meaning of the name "John" is the same as my own (basically, a very comical twist on the phonetics, which seems to be about right as most of us with that name are rather eccentric people). It comes from a Hebrew or Sumerian name meaning "Praise God (for this child)!" - taken to mean that the parents see this child as a gift or a blessing from God.
Given that I was born 2 months premature, it makes sense. Given no-name's behavior in Vietnam and role in covering up the existence of fully again as many POWs as were returned to us from Vietnam - who were never returned to us…. I can see why he is basically disowned of his name in its entirety.
But there seems to be more to it than that - a personal point that would pierce right through the traitor's skin. Q likes to make little quips that punch straight through to some insider information/joke/dealing. Like how Agent Smith tried to disarm Neo by mentioning his recent decision to help his landlady.
Which is why some around here are more than a little worried. The games of the blood clans are not clean, and even the most noble and well intentioned of them have reason to believe knowledge of their actions will not be well received by many.
Their fear is that a cannon has come loose near the powder-keg and/or that Q is simply playing for the position that has been held by the British royal family for a very long time.
That isn't what this is about, but you have to forgive ninja clans for not being very trusting of each other. Their history is almost nothing but stealthily killing each other from city-states that sit above our normal social infrastructure. Belief that any good can come from Q's plan is going to be met with quite a bit of resistance because being an optimist is how you get Diana'd.
America was formed by blood clans who decided that they wanted to live less like rulers and more as peers to each other. Rather than respecting this, a certain clan decided it would use its influence and proxies to subvert the processes and suppress the clans rooted in such an ideology.
So, blood lines are relevant, and more of us are members of them than we truly realize. Yet, we are only bringing them into relevance because law has evolved while man has been a bit more slow to do so, and it is necessary to claim the old rites long enough to counter the machinations of old. Our lines long disposed of the idea that humans were crops or substantially different from us.
If no-name were to go back on that idea, then it is understandable that many would consider that a betrayal of an order far more severe than a simple betrayal of country, but a betrayal of soul and legacy. It would be like having a child go and assist an alien species wipe out his own family and ideology. It's not a "I just think Russia/Britain would run this place better" - but "I am fully against the concepts you exist for and am willing to erase them, myself."