Honestly, I’ve heard so many stories about gedolim claiming Mashiach will come, and I never had any verification of them – except in once case. The one case was when R’ Elya Svei said that Mashiach would come in 2009. My brother was there and heard him say it in Darchei Torah. Obviously, Mashiach did not come in 2009.
I heard that R’ Kaduri zt”l was told by the Chofetz Chaim that he’d see Mashiach come. I heard that many Rabbonim said Mashiach would come in the year 2000. I’ve heard many stories where someone had a dream where a rosh yeshiva who had passed away came and told him Mashiach was coming that year. None of these actually came true (yet), so why would I believe that this happened?
Yes, we have to believe that Mashiach can come at any time. We can’t, however, live our lives assuming that it will. We need to prepare for both possibilities.
I also don’t believe this story because Eicha is still part of the Torah, and the Torah is eternal – do you think the Torah will cease to exist when Mashiach comes? Chas v’shalom!