"Did you know that it's possible to disagree with BOTH liberals and conservatives? It's called thinking for yourself!"
It's always vaguely annoyed me, but only recently have I been able to articulate exactly why. It's not that I am in any way opposed to the idea of thinking for oneself, because I consider that to be a very very good thing. Rather, it's that this particular expression of the idea is subtly framed in a way that tends to undermine that very ideal of independent thought.
First, it equates "thinking for yourself" with disagreement. In fact, it is entirely possible to think for yourself and decide that ultimately you agree with someone else; indeed, that's kind of what's supposed to happen if you're both applying valid reasoning processes to the same data. If, however, you choose to disagree with someone else's conclusion for fear of not adequately "thinking for yourself", you're falling into a particularly perverse form of dependent thinking: you didn't reach your own conclusion so much as you just embraced the opposite of whatever someone else concluded.
Second, it's also entirely acceptable and even appropriate sometimes to decide that someone else knows more about a particular subject than you do, and choose to defer to their expertise. I don't know an awful lot about medicine or human anatomy, but I had reason to believe that my surgeon knew what he was talking about when he said I had colon cancer, and so I made a conscious choice to follow his advice with respect to surgery and chemotherapy. Recognize that the decision to defer or not to defer to someone else's authority is always yours, and own that decision.
Third, look at the "BOTH liberals and conservatives" language, and how it subtly advances the insidious "both sides are just as bad" narrative, which I think is just monstrous. At best, it's a lazy way of seeming savvy, because it allows you to hint that you are aware of various sins on both sides without actually having to know any specifics whatsoever. But it has the effect of providing a karmic subsidy to whichever side actually is worse, at the expense of the side which isn't.
Do, by all means, think for yourself. And think about what it means to think for yourself, too.