The Arcane Trickster gains access to Wizard spells, primarily limited to the enchantment and illusion schools of magic. Four of the spells he knows can come from any school, however: one of his initial spells learned at 3rd level, and each of the ones learned at 8th, 14th, and 20th. This gives him the opportunity to have a spell in any school for one spell at each level he can cast. Further, like other casters who have known spells, the Trickster can change out one known spell when he levels up, with the following caveat:
Whenever you gain a level in this class, you can replace one of the wizard spells you know with another spell of your choice from the wizard spell list. The new spell must be of a level for which you have spell slots, and it must be an enchantment or illusion spell, unless you're replacing the spell you learned at 8th, 14th, or 20th level. (PHB pp 98)
This means that you can switch out your "any-school" spell of 2nd, 3rd or 4th level. However, your 1st level spell is not mentioned. Is this a simple oversight, or is there any reason why you might not be able to switch out the spell you learn at 3rd level?
This question applies equally to Eldritch Knight.
Best Answer
You can switch out any of the "any-school" spells
The PHB Errata, has now fully solved this issue.
From the "Rogue" section of the errata:
Similarly, the "Fighter" section of the errata states:
This means you can indeed retrain the "any-school" spell and replace it with another "any-school" spell as appropriate, both as an Eldritch Knight and an Arcane Trickster.
As a historical note, the fighter section was added to errata at least as early as 2016 while the rogue section was not fixed until 2020 though this tweet from Jeremy Crawford already stated that the same reasoning should apply: