Well, we wouldn't celebrate the harvest in June.
June is actually a pretty quiet month in the US as far as holidays go. Corpus Christi (tomorrow) would be the major Catholic church holiday that usually falls in June. Many churches have a procession with the Blessed Sacrament this day.
On the secular calendar there is nothing in June except for Flag Day (June 14), which is a minor patriotic holiday. People still have to work. In June the focus is on the summer beginning, the children have finished school until the fall. Independence Day (4th of July) is the next big holiday, where people have parades and fireworks and parties.
You should capitalize "Saint" when it is used as a title (Saint Patrick) and "Catholic" when it is referring to the church.