During meteorological summer, there is only one report of snow in Fargo Moorhead. On June 4, 1935 the observer in Moorhead wrote this in his notes that day:
A warning of light to heavy frost tonight if the sky clear was received this morning at 8:59 a.m. The warming was distributed by mail, radio, telephone and the newspapers. A few flakes of snow fell today at 8:23 a.m. and again from 12:07 p (sic) until 12:35 p.m. This is the first time since the station was established that snow has been observed in the month of June. The day was unusually cold for the the season.
Below is an image of the Daily Local Record of weather that day:
Too get temperatures cold enough for snow in early June, especially in the Red River Valley (Fargo elevation is just 900 feet above sea level) takes an almost perfect scenario. Based on the daily weather map produced that day (see below), it appears that the pattern was about what I had expected before I downloaded the data for that day. The set up was an intense area of low pressure centered just north of Lake Superior, this pulls down cold air not necessarily the arctic, but an area that is often even colder this time of year, Hudson Bay. The waters are usually just recently free of ice, or still partially ice covered this time of year. That advected some very cold air into the northern plains. Plus, high pressure was centered off to the west and southwest of the Red River Valley, that originated in Canada for another cold air source.
Plus, with such a strong area of low pressure north of Lake Superior, a strong cold pool aloft (500 mb and lower) was certainly centered over or near Fargo Moorhead that day. Considering the high was only 43° that day, I would estimate the temperature at 850 mb (roughly 4000 feet above Fargo Moorhead) at -4° C (about 25°F). That is pure speculation on my part, but to get flakes, especially over the noon hour, that close to the Summer Solstice, would take an usually cold air mass to produce snow flakes in the southern Red River Valley.
Other areas in North Dakota have recorded snow in June, especially the early part of the month. Dickinson for example had 1-4″ of snow on June 6, 2009 (see image below). Previous to that Dickinson recorded a measurable snow on June 1, 1951 and there is a reference to a snow event in June 1898. But most of the other June reports in North Dakota (and sporadic at that) were in locations with higher elevations in northern or western North Dakota.