<EDIT 2> Refer to my second post for a clearer explanation. I will leave my original post as a spoiler here.</EDIT 2>
<EDIT 3>For simplicity sake, let's ignore the island and think about something other than highest possible minimum. If it can be shown that "at least 1" is known by everyone, then the solution would be wrong even without figuring out some higher minimum. If we can first agree that "at least 1" is not new information, then we will at least know that the original solution is wrong.
Let's say that I am 1 of 20 people in a circle. I see 19 people wearing pants. I have no idea if I am wearing pants and I have no way of determining this.
1) Can I say with certainty that everyone else in the circle sees at least 1 pair of pants?
2) Can I say with certainty that everyone else in the circle knows that I see at least 1 pair of pants?
3) If #1 and #2 are true, then this logic can be applied to any individual in the circle, so it can be concluded that everyone sees at least 1 of pants based solely on my seeing 19 pairs of pants.
The solution relies on the fact that "at least 1 blue" is new information which triggers a cascade.
Wouldn't the entire population of the island be able to conclude that everyone else on the island knows there is at least 1 blue eyed individual already?
For example, every person on the island will see at least 99 blues and 99 browns. From this, they can assume that everyone else on the island can see at least 98 blues and 98 browns. Of course, the actual numbers will differ, but 98 is the lower limit for all perspectives.
A blue will see 99 blues and 100 browns, so he will assume that all other blues can see at least 98 and all browns can see at least 99 blues. Similar logic for a brown. </EDIT>
The rules of puzzle indicates that everyone is perfectly logical and is aware of the fact that everyone else is perfectly logical as well. Everyone also knows the same set of parameters, which means similar starting assumptions. This means that the choices made here will follow the superrational path rather than the rational path because the logic is shared while initial conditions differ by 1. Blue - 99 blue, 100 brown. Brown - 100 blue, 99 brown.
Short explanation -
Everyone already knows there is at least 1 blue-eyed individual on the island. Everyone also knows everyone else knows this because the minimum safe assumption will be 98/99 (depends on self color), based on superrationality. The guru stating there is at least 1 does not change anything.
Long explanation -
Even before the guru says there is at least one blue eyed person on the island, everyone on the island will have already made some logical conclusions. Ignoring guru for most of this.
1) A blue eyed person will have observed 99 blues, 100 browns and made some conclusions about what those people observe and what conclusions they can make.
2) A brown eyed person will have observed 100 blues, 99 browns and made some conclusions about what those people observe and what conclusions they can make.
3) All individuals know that they can be one of two states blue or not blue. They also know that everyone else thinks this as well. Everyone also knows everyone else knows everyone thinks this way. And so on. Works for any color.
4) Every person on the island can assume a lower bound for what every other person sees. It will be exactly the same thing but with 1 additional unknown. This is true regardless of color.
A brown knows that the 100 blue-eyed individuals he sees must see either 99 or 100 other blues(if he is blue). It would be impossible for any of the 100 blue-eyed individuals to see less than 99 or more than 100 blue eyed individuals.
Going with the lower bound, the brown knows that the 100 blues will all see at least 99 blue.
Those 100 blues will also see at least 99 browns as the original brown sees.
The 99 browns will see at least 100 blues.
Focusing on the blue seen. The brown can conclude that the 100 blues will conclude that there are at least 98 blues because the 100 see at least 99. Here is where the cascade begins for the rational solution. The brown can conclude that the 100 blues will conclude that there are at least 99 blues and will conclude that those 99 will see at least 98 and those 98 will see at least 97... all the way down to none.
However, being superrational beings, and knowing about each other's superrationality, they can conclude that the blues will see at least 98 and assume everyone else will see at least 98. This is because the brown will consider the blue's reference point and knows the blue will consider his as well.
5) Brown knows everyone he sees will see at least 98 browns and 99 blues with self unknown and the original brown unknown. Since we consider the least case, it will be 98 browns, 99 blues, self unknown, 1 not blue.
To illustrate the point, we must now consider what a blue would see and assume about everyone else which everyone on the island will do as well.
A blue knows that the 99 blue-eyed individuals he sees must see either 98 or 99 other blues(if he is blue).
The blue knows that those 99 blues see at least 98 blues. This will cascade to zero as well. However, the blue knows that all of the 99 blues he sees are superrational. They will consider themselves as stateless and perform the same analysis with 1 less blue and that every other blue will think exactly as he thinks and see at least 98 other blues.
The blue knows a brown will see 99 blues and assume exactly as he has.
6) Blue knows everyone he sees will see at least 99 browns and 98 blues with self unknown and the original blue unknown. Since we consider the least case, it will be 99 browns, 98 blues, self unknown, 1 not blue.\\
The minimum information required to trigger a cascade for blues would be 98 blues.
We can consider what happens if the guru says "at least 97 blues".
For the brown, he already knows this. He knows that of the 100 blues he sees, they also know this because they can't see less than 99. He also knows that the 100 blues knows that the people they see will see at least 99 blues as well. This is because of  which means between any two individuals, there is only one additional unknown.
The blue also knows that of the 99 blues he sees, they will see at least 98, and knows the browns he sees will see at least 99.
I have no idea if this is logically sound or not. Superrationality on its own is already confusing because there is a giant loop of logical deductions. Instead of maximizing their gain, I am just using the fact that they know everyone else can reach the same conclusion as them to stop the cascade because no one on the island can see less than 98 blues or 98 browns from any perspective. This can be arrived at from every perspective as well.