Nope, sorry. Pokemon only learn attacks if:
a) IT learns it naturally, not it's evolved forms (sorry, no Body Slamming Magikarp)
b) IT can learn the attack by TM (as I mentioned, IT, not it's EVOLVED form)
c) It can learnt the attack as a breed-only attack.
Since Eeeve can't get Flamethrower naturally, by TM, and the attack is not breed-only, Eevee can't learn it.
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