I think they have every right to say it, but it's counter-productive to their cause.
Many ideas that are common knowledge in social justice circles sound ridiculous to people not used to them (sometimes because people lack context or haven't heard the arguments for them, sometimes because they actually are ridiculous). When people encounter an idea that sounds strange because it's outside their usual frame of reference, and they ask you to explain it, you are being given a golden opportunity. You have someone who has not yet made up their mind on something asking you to give them the first impression on it.
Ideas have a huge first mover advantage. People don't change their minds easily; once you believe something, the natural inclination is to continue to believe it and you will be more sceptical of anything (and anyone) that disagrees. And so, convincing someone who has no opinion on the matter is much easier than convincing someone who already disagrees with you. You can present the most favourable arguments and evidence, you can frame it in the way most convenient to you, etc.
The problem being, of course, that not everyone who asks questions is genuinely curious. People who have already made their minds up can still pretend to be "just asking questions". (They used to call it "JAQing off", not sure if that's still in the Official Social Justice Lexicon). And so social justice people get soured on those assholes and assume everyone who doesn't know that thing they think is obvious* is a troll being deliberately obtuse.
Which doesn't change my original point. It's a waste of a good opportunity to get people to take you seriously, rather than being annoyed that you refuse to explain and looking up the first hit on google, which can as often as not be from your ideological opponents. Does it mean you occasionally have to deal with assholes? Yes, and that sucks. If you don't want to deal with another asshole on the off-chance it's a case of genuine curiosity, it's your trade-off to make.
*and if you hang out with one group enough, it gets harder to see why its memes are obvious to you but not to other people.
There's another that annoys me even more. Your average conservative, when challenged to provide a citation for his claims, will typically demand that you provide that citation (the phrase used is "I won't do you research for you"). How do people respond to this fallacy? For me, it's typically just screaming.
Depends. If I'm feeling especially epistemologically virtuous, I might just do the research myself and see if I find anything interesting. Argue to learn and all that. If not, I'll politely inform them that refusing to present evidence is not the way to convince people. Should that fail, I'll declare the argument a waste of time and leave. Might or might not tell them to fuck off, depending on my mood.