I have found a catch in the solution of this logic puzzle. Provided that you are not allowed to give the prisoners any new information, the statement "at least 99 of you have green eyes" does give them new information. To give 'new information' can be defined as when the statement you say makes the prisoners immediately know about something which they previously did not know about. To prove my point let's shrink the puzzle down to just 2 prisoners Prisoner A and Prisoner B. Imagine the point of view of prisoner A, he would have made a list of what he knows and what he doesn't know.
Point that A knows and is completely certain of- Prisoner B he sees have green eyes (i.e. at least one of them have green eyes)
Point that A doesn't know and is completely uncertain of - Whether prisoner B sees green eyes (i.e. uncertain about B knows that at least one of them have or not) this is because if in case A have non green eyes B would not know that at least one of them green eyes.
Now when you tell them at least one of them have green eyes A will be immediately sure that B knows at least one of them have green eyes as you said the same. Hence A will gain new information immediately after you gave your statement as before your statement he had no information about whether B knows at least one of them gave green eyes. This disproves the point at 3:35 (Timestamp in the YouTube video) that the new information was not contained in your statement itself as it indeed did contain new information. Looking forward for someone to support/oppose my argument.
Edit-I just realised I have been opposing the solution by supporting the very thing the solution is trying to suggest. If you listen carefully the video from 3:35 to 3:40, they mention that "The new information was not contained in your statement itself but in telling it to everyone". If you say atleast one of you have green eyes to everyone privately there's no new information gained by anyone as the prisoner is still not sure of whether the other prisoner knows atleast one of them have green eyes. He will be sure about it only when he knows the same statment was made to the fellow prisoner. Which is the same thing they are mentioning in the solution. Also one of the answer saying that when they were publically told some statement adds new information to them is correct, as the fact comprehended by every prisoner that 'the statment was told to everyone in the same room/auditorium and is know known by everyone' forms the basis of the inductive reasoning performed here.