Cool pattern on that rock! Thanks for the pictures and Welcome to the Board, Mishihari. 
I believe you have found a fossilized piece of coral. Not Petoskey Stone, but some other type of coral. Some Petoskey stones will fluoresce, but not all of them. Besides for the uniform rectangular spaces the light blue color of this piece is interesting. Most of the coral fossils up there are cemented with a Calcite base, which usually fluoresces red, pink, to yellow, but sometimes blue. And there could be traces of other minerals that cause the blue color.

Here's a link to list of minerals and the different colors they can fluoresce under long and short wave lights.
http://www.galleries.com/minerals/property/fluotabl.htm

Your fossil coral pattern is a view from the side of the coral and not the top. I don't know much about Corals, but they don't all have that type of segmented growth, so that should help in identification.
In a quick search I did find one type that looks similar,
a Bryozoan type,Trepostomata, named  "Dekayella Praenuntia Ulrich" which is/was found in Minnesota.  These typically are older corals dating to the Ordivician.
Some others on here may be able to provide better information but Nice Find, Good Eye, and thanks for the pictures! 

Thanks, Dave. I knew you could do it!
Charlevoix Stones sounds pretty impressive!

Thanks!  I thought it was probably a fossil, but I had no idea what type.  I found it at Black River Harbor Beach in Michigan, in case anyone is interested in looking for one of their own.