No eclipsing by the earth. It'll be in a wide halo orbit around ESL2, not at the point itself. Due to complicated math, being in a halo or Lissajous orbit requires less stationkeeping than trying to balance at the point itself.
ESL2 is only chosen as an orbit to ensure that the Sun, Earth, and the Moon are all on the same side of the telescope so that the sunshield can protect the telescope against thermal radiation from those as well in order to keep the telescope cool.
Lissajous orbits which are not halo orbits would lead to occasional partial eclipses, which is actually not desirable since the telescope is solar powered, and also you want it to go through as few unusual events as possible.