No it doesn't work. The Mystic Theurge isn't an Arcane Class, it's an [Arcane AND Divine] class.
Your player is reading it like this:
This essentially means that he adds the level of mystic theurge to the level of whatever ...
... other arcane spellcasting class ...
... and ...
... divine spellcasting class ...
... the character has, then determines spells per day, spells known, and caster level accordingly.
But it should actually read like this:
This essentially means that he adds the level of mystic theurge to the level of whatever other ...
... arcane spellcasting class and divine spellcasting class ...
... the character has, then determines spells per day, spells known, and caster level accordingly.
The word "other" applies to both the subjects that it precedes.
I have seen people joke about using Evangelist to continue prestige classes, but I have never seen it used that way in any build(s) people have posted [that I paid attention to, anyways].
And I have personally combined/gestalted Arcane Archer and Eldritch Knight into one full BAB, full 10/10 spell progression, and all/both their respective class features... individually, neither prestige class is worth giving up what you already have going... combined, though, it's not TOO bad... single class Myrmidarch is still better, but whatever.
As for your question, it's pure BS cheese... and, at best, I would allow them to continue the arcane spellcasting. But no, the divine spell progression would not piggyback just because Mystic Theurge does. Evangelist would continue both because Aligned Class continues class features. Eldritch Knight continues spell arcane progression... that is it.
Yeah have a look at the EVANGELIST and the MORTAL USHER. These are - as far as I know - the only 2 prestige classes that advance ALL class features, so they could potentially be used to advance Mystic Theurge. There is some debate to this though; they don't specifically day they can't advance prestige classes, but they don't specifically say they can either . If you think it'd be unbalanced you're allowed to say no.
Regarding this player: This sounds like they're trying to "Win" Pathfinder. They're also doing so by looking for loopholes. I understand the desire to make the most of something but if they're trying to pull one over on you then that's dishonest behaviour.
If you just came here because you weren't sure then "no harm, no foul", but if you came here because your player wouldn't accept your ruling then you have a problem player. As the GM you're allowed to interpret the rules as you see fit, provided you're honest and up-front about it.
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When trying to advance spellcasting with a prestige class, you can only advance the spellcasting of a class that granted spellcasting… this doesn’t outright prevent choosing a prestige class to advance though, but it does make most spellcasting prestige classes invalid choices. There are a few prestige classes that actually have their own spellcasting progression that can be advanced by another prestige class.
So, no they can’t advance Mystic Theurge spells per day with Eldritch Knight… as for equipment trick… some people insist it works, but nothing about the trick actually grants you the ability to cast higher level spells. You use an extra material component to treat a spell as if it were one level higher AFTER you cast it… in other words, you still cast a 1st level spell, but that first level spell is treated as a 2nd level after the fact… the only “early entry” trick that actually works within the rules is Faith Magic, and it actually doesn’t even get you in until one or two levels after you could have gotten in normally… or if you have a GM who takes a generous reading approach to the Medium class you could get in with just 7 levels in one class… though it’s be a terrible way to play mystic theurge…
it might work the other way around where the the MT increase the knight's arcane caster level. but he probably want to level as the knight for bab reasons..
look at it this way.
you got levels in an x divine class and in a Y arcane class.
MT increase levels for both X and Y. eldritch knight only for Y
they are both only mirroring the original. saying 'go back to x and\or y and increase it by 1'. in a way they don't really have a spell caster level of their own.
let's say the faq about using spell like abilities for requirements wasn't published and some1 got in via an arcane sla, at prestige class 4 what spells per day would he have? if it was an arcane casting class there should have been a table to show it's progression, by itself it's not an arcane caster class, it's a booster to one.
think of them as a magical enhancement to a weapon. they both enhance the weapon. not each other. without a weapon they fail to stand alone.