The National Centers for Environmental Information (formerly NCDC) released their numbers for June, showing that Illinois did indeed have its wettest June on record with 9.30 inches (according to their calculations). That made Illinois the wettest state** in the US for the month.
Here are the national maps showing the statewide precipitation values for June 2015. While Illinois received 9.3 inches of precipitation, California received only 0.23 inches – an amount 40 times less than Illinois.
Here are the rankings by state and the departures from the 20th century average. The scale goes from 1 being the driest and 121 being the wettest. Ohio and Indiana also had their wettest June on record as well.
Here is a closer looking at the precipitation rankings by climate division across the US. The climate divisions (or crop reporting districts) in Illinois were all ranked as either the wettest (121), second wettest (120), or third wettest (119) on record. One of the outstanding features of this June was how widespread the wet conditions were in Illinois. It is more typical in other years to have one part of the state with very wet conditions while another part may be close to average (such as, 1993).
The next figure is the plot of June precipitation in Illinois from 1895 to present with this June clearly standing head and shoulders above any previous record. Seven out of the last eight June totals have been above the 20th century average of 4.09 inches.
**There are no statewide records for Hawaii. However, an examination of the four main sites in Hawaii indicate June totals that are far less than 9.3 inches.