No. At the moment, Christians are to judge those in the church (1 Cor. 5:12; cf. 1 Cor. 6:5) and, in the future, they will ‘judge the world’ (1 Cor. 6:2) and even “judge angels” (1 Cor. 6:3). But, for the time being, Christians are not to judge those outside the church: they are to leave that to God. As Paul says:
…what have I to do with judging outsiders? Is it not those inside the church whom you are to judge? God judges those outside…. [1 Cor. 5:12-13]
Of course, the most important “judging” we are to do is of ourselves because, ‘if we judged ourselves truly, we would not be judged’ (1 Cor. 11:31).
I think a lot of the answers on this site are quite involved, intelligent, and complex. This answer, however, is amazingly in poor taste. It is extremely legalistic and does not derive implications from the text. Rather it assumes a very literal translation. This in my opinion is the type of interpretation that is very damaging to the church and the meaning of God’s word. Quite surprised to see an answer like this on this website.
Personally, I think that a clear line is drawn between affairs “inside church” and affairs “outside church” in 1 Corinthians 5 & 6, and so I would have taken those passages in exactly the same way as this answer did. Perhaps the answer then derives the “no jury service” from this principle solely by implication, but I can’t see how the interpretation doesn’t derive implications from the text or how it assumes a very literal translation. Maybe I’m just biased by already agreeing with the author, but could you possibly spell out what you mean in a little more detail?