I see more of that style of speech (from both sides of the political spectrum) in social media and news site comment sections than I have heard on conservative talk radio. I don't listen to it much anymore, but I used to turn it on now and then because my mom (who was ill and staying with me) tired of NPR, so I'd switch back and forth.
I don't remember hearing any of the hosts we listened to using the term 'snowflake', not much on anyone being 'triggered' (I've mainly seen that mentioned in liberal leaning media, actually -- 'trigger warnings' and the like), but I've seen 'snowflake' used a gazillion times on social media and news sites' comments and any other online forms of dialog. A few of the hosts talked about 'micro-aggression', but that was when they were reading news reports (micro-aggressions and 'triggering' are apparently related).
Rush might have used terms like that, but I never listened to him.
Never personally heard any talk host talk about leftists needing safe spaces. If they said it, I don't remember hearing it, so it couldn't have been much. Maybe Michael Savage would have, it sounds more like something he would have said. But I think his network dropped him, he's not on the air anymore, and he had lower ratings than other hosts. He seemed to be the most extreme of the right wing hosts, Mike Malloy being the most outspoken and opinionated of the progressive talkers (when he was on the air).