For impenitent sinners, one of the most convenient verses in the entire Bible is Matthew 7:1, in which Jesus says "“Judge not, that you be not judged." The least convenient verses, however, follow this immediately: "For with what judgment you judge, you will be judged; and with the measure you use, it will be measured back to you. And why do you look at the speck in your brother’s eye, but do not consider the plank in your own eye? Or how can you say to your brother, ‘Let me remove the speck from your eye’; and look, a plank is in your own eye? Hypocrite! First remove the plank from your own eye, and then you will see clearly to remove the speck from your brother’s eye. “Do not give what is holy to the dogs; nor cast your pearls before swine, lest they trample them under their feet, and turn and tear you in pieces." (v2-6).
From the context, we see that Christians are not forbidden from making any and all judgements. Indeed, in order to cease giving out corrective advice, we first have to judge that the people we are no longer judging are dogs and swine. Now, which would you rather be judged as? Wise enough to accept correction, or scummy enough to be unworthy to receive it?
This leaves us with the immediately answered question "then what does Jesus mean when He says judge not?" Put simply, it is a condemnation of hypocritical judgement, i.e. judging others for what you, yourself, are guilty of. Furthermore, if you show love and mercy in your judgments, you are given love and mercy in return. Is the purpose of your judgement reconciliation? Then when Jesus judges you, it will be for reasons of reconciliation. Is it correction? Then Jesus will judge you to correct you. Only if your motives are wrong is your judgement wrong.
Ironically, this precludes the erroneous interpretation of Matthew 7:1. After all, anyone who tells you "judge not" is judging you. They have seen you judging, they have judged that you are wrong to do so, and they have sought to actively stop you from doing so. But in doing so, they not only become judges, but it is 100% impossible for them to do so without a plank in their own eye! Why? Because they are doing the very thing they have judged it is evil to do!
Because of this, you "win" by default. Of course, the purpose should be correction and reconciliation, not winning an argument, but you do win the argument. Whether this leads to reconciliation and correction is entirely up to the person in question. If they are consistent, there are two ways it can go. On the one hand, they may keep their wrong interpretation of Matthew 7:1, but apply it consistently, realising that they no longer have the right to judge you, and therefore, they cease to be obstructive.
Alternatively, they may realise there are things it is entirely legitimate to judge, and that Scripture repeatedly tells us to do so. We are told not to judge by the appearance, but judge righteous judgement (John 7:24), to expose the unfruitful works of darkness (Ephesians 5:11), to mark those who cause division and spread contrary doctrine (Romans 16:17), test all things and hold on to the good (1 Thessalonians 5:21), that we will judge things that pertain to this life as we will eventually judge angels (1 Corinthians 6:3), and a host of other things. There is so much in the Bible that tells Christians to judge all things (1 Corinthians 2:15) that if anyone even hints at Matthew 7:1 condemning all forms of judgement, one might rightly call him a fool.
Sadly, human nature means neither positive outcome is guaranteed. The hypocrites who so abuse Matthew 7:1 are usually doing something worth judgement, or know someone who is, and, being hypocrites, may very well continue to abuse Matthew 7:1 even knowing it does not mean what they want it to mean. But you don't have to listen to them, for whatever speck is in your eye, you don't want their plank going anywhere near it. Therefore, there are only 3 possibilities here:
They receive correction, and you do not have to slow down your walk; they will even join you.
They will remain in error, but not in hypocrisy, so you do not have to slow down your walk, and they will shut up and stop trying to impede you.
They will remain in error and hypocrisy, and you do not have to slow down your walk. They will continue to squawk, but the one who is in you is greater than the one who is in them.
In all cases, God's children should heed Matthew 7, as we should heed all God's words: in context. Do not judge hypocritically, but if you may judge without hypocrisy, do so with love, mercy, and righteousness. It's that easy.