Is there any real substantial evidence that Rav Yonasan Eibshitz was a Sabbatean? On what basis did Rav Yaakov Emden hound him?
There were many proofs put forth that Rav Yonasan Eibeshitz zt"l was a Sabbatean, published in sefarim such as Sefat Emet (anonymous, often attributed to Rav Yaakov Emden, but recently published materials suggest Nechemiah Reischer, a disenchanted R' Eibeshitz enthusiast who became an admirer of R' Emden). One such proof, as described in this article by Rabbi Dr. Leimann, is as follows:
Rav Yonasan Eibeshitz engaged in practical kabbalah, writing amulets, with the instruction that people should not open them. He wrote such an amulet to a woman, but it did not work; she died. The husband gave the amulet over to Rav Yaakov Emden who was also a master of kabbalah. He read it and saw Sabbatean messages in it, such as "Yeled Yulad Lanu, Ben Nitan Lanu" combined with Shabtai. Rav Emden published this in the sefer Sefat Emet.
Rav Yonasan Eibeshitz's response to this was that they were misreading the amulet. E.g., what Rav Emden said was a tav was really a chet, and so on.
What the article then documents is that Rav Yaakov Emden went to a French court together with two of Rav Yonasan Eibeshitz's talmidim. The French court copied over the amulet and the two talmidim signed an admission that this was the accurate text of the amulet.
Since Rav Yonasan Eibeshitz's response was shown to be false, and without that explanation, the amulets are properly understood as Sabbatean, this is evidence that he was a closet Sabbatean.