According to D&D 5e they shouldn't get it. But BG3 isn't really D&D 5e so it might be possible in-game at present. I don't see why they should be denied access to such a spell though. An alternative nearing launch could be to multiclass or to grab the magic initiate feat for the spell.
This is incorrect. Eldritch Knights largely learn spells from the evocation and abjuration schools, but can learn a few spells from any school.
You know three 1st-level wizard spells of your choice, two of which you must choose from the abjuration and evocation spells on the wizard spell list.
The spells you learn at 8th, 14th, and 20th level can come from any school of magic.
Taking the magic initiate feat or an extra level of wizard will give you more low level spells and cantrips, though.