Since Greedo specifies that he works the first time you're attacked and the first time you defend, am I reading it correctly that he has no negative effect that round if you manage to evade that initial attack?

yes, I believe you are right. It says first time you defend, not first time you recieve damage. So if you evade the attack or if it only reduces shields, then no damage cards are dealt, and you do not suffer the effect.

On greedo, is the attack and defense referenced on the card associated with any ship on my squad or only the greedo equipped ship? Apologies if this has an obvious answer, I'm just a little unclear on the wording.

On greedo, is the attack and defense referenced on the card associated with any ship on my squad or only the greedo equipped ship? Apologies if this has an obvious answer, I'm just a little unclear on the wording.

Only the Greedo equipped card.

When cards reference "you", they mean the ship/pilot (not the player).

I have another Greedo question:

If you do another critical hit with the Greedo'ed ship, does that effectively negate his card text? He makes the first card deal face up... crits are dealt before hits... natural crits are already dealt face up.

You are mistaken on the order damage is dealt. Normal damage is always dealt before Critical Damage(Rulebook page 13). So the only time Greedos ability would have no effect is if your first attack dealt only Critical Damage