When we write the date we always write the day first:
7th June 2018, or 7/6/2018
And when spoken we normally say the day first:
The seventh of June.
However there are some exceptions where it can be normal to hear them spoken the other way around. When we do though, we nearly always prefix the day with "the" and include the ordinal indicator, which I notice Americans do not. It seems to be more common when referring to specific dates, such as holidays, eg:
"November the 5th" (Guy Fawkes night in the UK)
"December the 25th" (Christmas day)
To be clear these are not exceptions to any rule, just occasions where I, as a native British English speaker have noted that for some reason some quote the month first. You can see Guy Fawkes night rendered as "November the fifth" in this article from British newspaper The Guardian and also in this Oxford reference article.
If I was to guess at a reason why we normally say the day first it would be that this is considered to be the most pertinent part of the date. It is often assumed that one knows which month it is, and when someone asks for the date it is not unusual for the reply to simply be "the sixth!"