These represent the members Reputation Level.
When you use the sites Rating function and agree, then the member in question receives one point.
For each 200 points you get an extra greenie.
I want to point out the green squares are not a reliable guide to determining whose advice to follow. We have some great people here giving reliable answers to tough technical questions in one area they know well such as Northern Heat and tkrussel who only have 1-2 boxes. You have other people with a whole string boxes mostly earned by displaying their concern in some areas. How many boxes you have is very subjective. And the boxes you earn in one forum, show up in all of them. You need to carefully weigh all the answers you got here and give others a chance to point out flawed ones.
In Dogs, I have seen some terrible answers Googled up by people with plenty of green boxes.