In the Northern Hemisphere, the longest day always occurs at the summer solstice (June 20 or June 21). But for the days of shortest shadow, one must make a distinction between locations the north of the Tropic of Cancer and locations between the Tropic of Cancer and the equator.
For places to the north of the Tropic of Cancer, the summer solstice is also the day when the Sun casts the shortest shadow around noon. Indeed, every day around noon, the sun passes to the south of the local zenith. At the summer solstice, the position of the Sun at noon gets the closest to the local zenith, so then it casts the shortest shadow.
For places at of the Tropic of Cancer, the Sun passes (almost) directly over the local zenith at the summer solstice, so it casts no shadow at noon. So for these locations, the summer solstice is also a zero shadow day.
But for locations between the equator and the Tropic of Cancer, the sun can pass the zenith on the north side during summer. This means that for these locations there are two zero shadow days, which do not coincide with the summer solstice.
The image below illustrates this: it shows the daily motion of the Sun at several times of the year, for a location with latitude $\varphi = 11.72^\circ$, which is halfway between the equator and the Tropic of Cancer.
The declination of the Sun is denoted as $\delta_\odot$. Since the axial tilt of the Earth is $23.44^\circ$, the summer solstice corresponds with $\delta_\odot = 23.44^\circ$ and the winter solstice corresponds with $\delta_\odot = -23.44^\circ$. The equinoxes are the dates when $\delta_\odot = 0^\circ$. Note that at the summer solstice the Sun passes the local zenith $Z$ on the north side at noon.
The zero shadow days are the days when the Sun passes through the local zenith. This occurs twice a year, when $\delta_\odot = \varphi$ (so in this case when $\delta_\odot = 11.72^\circ$). It's straightforward to see why, because the Sun lies on the celestial equator when $\delta_\odot = 0^\circ$, which is perpendicular to the celestial north pole $P$, and the angle between $P$ and $Z$ is $90^\circ - \varphi$. A quick look at an ephemeris table tells me that the corresponding zero shadow dates for this location are around April 21 and August 22.