Question: Does a potential defendant have to come to beit din if he knows he is right? Does he have to help clarify the story, which might work to his detriment, if the plaintiff has no proof? Don’t we say hamotzi meichaveiro alav ha’re’aya (the claimant has the burden to prove his case)?

Answer: Ostensibly, one who is positive (and is in a position to know ) that he does not owe money is not required to volunteer to go to beit din to prove it to the other side. After all, in certain cases (see complicated halachot in Shulchan Aruch, Choshen Mishpat 4) one may "take the law into his own hand" without resorting to the hassle of adjudication. Likewise, if one knows he does not owe money, he may make things difficult with the hope that the other side will give up and/or agree to a compromise (Pitchei Teshuva CM 12:8; Tumim 12:5).
However, this makes more sense if the other side is aware of the truth, not if he sincere but is making a legal or factual mistake. Furthermore, the formula is different in regard to response to a beit din which summons the person while operating within its authority. Its instructions must be followed, as we see from the fact that if one does not do so, the halacha is that he is to be put in cherem (Shulchan Aruch, CM 11). The fact that many batei din (including ours) do not resort to charamim does not mean that failing to conform to the instructions of a neutral, authorized (by halacha) party is tolerable.
A litigant may certainly raise issues (e.g., wanting to go to a different beit din – Shulchan Aruch, CM 14; asking for an itemized claim – Shach 11:1; requesting time to work out a compromise), but he should do so appropriately. A defendant’s claim that he need not come because the plaintiff has no proof is invalid. First of all, only time will tell. Secondly, a litigant has the right to make the other side deflect his claims and arguments when interrogated before and by beit din, which is harder than denying a claim one-on-one. Third, logical claims may convince beit din to employ compromise, including in cases where oaths, which we no longer administer, are called for.
We now move on to proceedings within beit din. If a litigant possesses information that is valuable to the other side, beit din is required to prevail upon him to produce it (Shulchan Aruch, CM 16:3). He must not say: "You did not prove your case, so you lose." The damaging information does not always mean that the one who admits or hands over documents should or will lose the case. In fact, sometimes he will be able to use a migo, i.e., "since I was honest enough to give up a false but potentially winning claim, believe me regarding a different claim without further proof." (The complicated parameters of this rule are discussed in CM 82). The litigant is not required to produce a document that only looks relevant but is not for a reason he cannot prove (see Tumim 16:2).)
In limited cases, there are opinions that it is permitted to make claims that are not true but are equivalent to the truth (see Mishpat Aruch 75:1:30-34). However, the general rule is that litigants must tell everything as it is, and let the dayanim decide (see Shach, CM 75:1). It is not valid to rationalize that "since the other side is lying or exaggerating about certain things, I can do the same." This is all the more so considering that often (based on my experience) litigants who were convinced they were right were not. (Mistakes can be attributed to a lack of halachic knowledge in often complicated matters and the natural tendency to see things in a way that advances one’s interests.) Thus, one rarely knows when there are righteous ends that might justify certain means. Finally, a warning – dayanim often uncover or sense a litigant’s attempt to "improve" his claims, which weakens his case.
We pray that litigants will cooperatively and truthfully join dayanim in searching for a correct ruling, which is Hashem’s will.