New South Wales
Would you have to perform an evade when pushing a monster?
Yes, you do. Before doing any part of the Push action, you must evade the monster, as it an action which is neither a move nor an attack.
I mean, you're already making a brawn check. Seems crazy to have to evade a monster first in order to test your strength against them.
Rules will often have some weird interactions, this is one of them. Think about it as the monster taking a swipe at you before you can get close enough to grapple it.