Let's say I have 6 hull and 2 shields. I get hit with 2 crits and 2 hits.

Do the crits hit on the shields first or do the hits first cancel shields then let the crits eat the merry hull?

Unless an ability says otherwise, you cancel/apply regular hits before critical hits.

In your example, your 2 shields would be removed by the 2 regular hits, and then you'd deal 2 faceup damage cards to your unshielded hull.

I think the only time you can cancel [crit] results before [hit] results are when an Autoblaster is used which also happens to make those [hit] results uncancellable.

When it is time to SUFFER damage I believe the [hit] results are always resolved first before any [crit] results thereby maximizing the chance that a face-up damage card is dealt.

Addition - An Example:

Ten w/ Autoblaster shoots a cloaked Phantom that has lost a point of shield from something.

After roll modifications Ten gets [boom] [kaboom] [kaboom] with attack.

Phantom rolls two [evade] results.

In the Compare step the [kaboom] can be cancelled first but Ten says one [kaboom] can not be cancelled. This means that a [kaboom] and [boom] go through.

Now when it is time for the Phantom to SUFFER the damage the [boom] moves back to the front of the line and takes out the remaining shield token. This allows the [kaboom] to deal a face-up card to the Phantom; of course it happens to be a Direct Hit and the Phantom goes POOF! Had the [kaboom] been suffered first it would have been absorbed by the shield and then the [boom] would have just dealt a face-down card leaving the Phantom alive.