The standard installation finished height for Wainscoting is a 36" height for an 8 foot ceiling and approximately 42" for a 9 foot ceiling. In this case the ceiling is higher and the wainscotting is done on the height that matches with the wall behind the pillar...36".
As for the squares they are 3"6" with 8 " space between.
However the squares measurements are determined by the wall lenth and can be quite compicated. Here is a website that may help as well as a few pictures that will show how different wall lenths mean different squares.
http://www.garymkatz.com/TrimTechniques/Wainscoting/WainscotingArticle.htm

Thank you so much for the helpful and informative answer!!!