If we look at Milankovitch cycles we can see that equinoxes will be 1/4 of the Year exactly once every 500 years, as then then the movement will be a circle.https://en.wikipedia.org/wiki/Milankovitch_cycles
And then the Earth will not move faster when its closest to the Sun, as the distance will be equal whole year.
So will day and night be equal... Again depends on our definition of day and night. Looking at the light, of course he is right, depending where we are there are a lot more factors involved.
On the day of an equinox, daytime and nighttime are of approximately equal duration all over the planet. They are not exactly equal, however, due to the angular size of the Sun, atmospheric refraction, and the rapidly changing duration of the length of day that occurs at most latitudes around the equinoxes. The word is derived from the Latin aequinoctium, from aequus (equal) and nox (genitive noctis) (night).
And again we get into apparent(what is seen from Earth) vs mean(what would be seen from outside the Earth in this case).
As both hemispheres will be equally illuminated:
The equinoxes are the only times when the solar terminator (the "edge" between night and day) is perpendicular to the equator. As a result, the northern and southern hemispheres are equally illuminated.
However there are other factors that may change the light and how it falls.
Overall in the mean values/view its a important moment, however if we go into apparent values its a mess there, as looking from Earth there are many factors that may change the impression(logically even a very, very big lamp can do it).
Conclusion - use mean values people!The rest is unreliable!(a joke, in practice all seems to be unreliable anyway).
One thing helpful for understanding it can be this:https://www.amusingplanet.com/2017/04/lahaina-noon-when-shadows-disappear.html