I've been toying around with a build idea, it features Bossk in his ship, Greedo, and the Mangler cannon, but before I do much else with it I want to be sure I am right about how this interacts.

Say Bossk is rolling his normal 3 attack dice, he gets 2 hits and a crit. His ability triggers (first?) which converts the crit to 2 hits. He now has 4 hits. For the sake of simplicity the defender has no way of canceling damage through dice or evades in this scenario and the target is unshielded. We got a Bossk with 4 hits, the Manger cannon effect now happens (second?) which turns 1 hit into a crit so we end up with 3 hits and 1 crit. Damage is dealt and Greedo activates (third?). The first damage is dealt face up, turning this into 2 crits and 3 hits.

Is this the right way to interpret the rules here with the order of cards activating?

Nnnnot quite.

The order of the attack is as follows:

1. Attack is declared.
2. Attacker rolls Attack Dice
3. Defender modifies Attack Dice
4. Attacker modifies Attack Dice < Mangler Cannon happens here
5. Defender rolls Defense Dice
6. Attacker modifies Defense Dice
7. Defender modifies Defense Dice
8. Defender cancels [boom] results
9. Defender cancels [KABLAM] results
10. If any uncanceled [boom] or [KABLAM] results, then Defender is Hit < Bossk happens here
11. Defender discards Shield tokens to cancel [boom] results
12. Defender discards Shield tokens to cancel [KABLAM] results
13. Defender receives facedown damage cards equal to remaining [boom] results < Greedo happens here
14. Defender receives faceup damage cards equal to remaining [KABLAM] results.

DraconPyrothayan is correct, so to follow through your example.

Bossk rolls 2 hits and a crit.

Defender would use any effects here (R7, Sensor Jammer etc) - lets say he has nothing.

You would use mangler here, you now have 1 hit, 2 crits

*Defender rolls defence dice and rolls 3 evade.

You have nothing to modify the dice and they choose not to.

They cancel all results and the attack misses.

end of attack.

*If they rolled evade, focus, blank

you have nothing to modify the dice, they spend their focus and now have 2 evade

defender cancels dice and you are left with 1 crit

You would now activate Bossk and change that 1 crit into 2 hit results.

The defender then suffers 2 damage, the first one of which is face up from Greedo.

end of attack.