If an Eldritch Knight focuses on spells that don't have an attack roll or saving throw, INT (spellcasting stat) becomes less critical. But is there still a minimum INT required to cast spells of a certain level? How low of an INT can an Eldritch Knight get away with?
But is there still a minimum INT required to cast spells of a certain level?
No. Your spellcasting ability is only used to determine spell save DC and spell attack roll bonus.
How low of an INT can an Eldritch Knight get away with?
There's no minimum, rules-as-written. Even spells like feeblemind now explicitly state that spellcasting is blocked rather than relying on low ability scores to prevent casting.
Rules-as-written, you can cast spells with an Int of 8 if you're using the default array. Your ineptitude at doing so is only reflected in the attack modifier of -1 + proficiency bonus and extremely low spell save DC of 7 + your proficiency bonus.
If you're rolling ability scores, you could conceivably have an Int of 3. A Wizard could with an Int of 3 could still cast 9th level spells. They just wouldn't be very potent as far as attack rolls (+2 modifier) or save DCs (DC 10).
A low inteligence Eldritch Knight is not only possible, but viable. There are plenty of spells you could use that don't need spell dc or attack rolls. even if you look at just evocation and abjuration (mostly abjuration). These spells are Shield, Magic Missile, Protection from good and Evil, Darkness, Sending, Remove Curse, Alarm, Arcane Lock, Daylight, Fire Shield, Dispel Magic (til 4th level), Counterspell (til 4th level), etc and plenty more outside those school. Not to mention there are plenty of useful cantrips which have no DC requirement. True strike, blade ward, light, mending,etc.
As you don't rely on Int for anything but Saves and Spell Attack Rolls, if you avoid spells with those, you don't need to have any given int score. Go ahead and ignore it for better stats for your build.