Per COC RULE 24 (B) (4) (c) it's a Force Majeure Event, which UA will rebook or credit you for:
In the event of a Force Majeure Event, UA without notice, may cancel, terminate, divert, postpone, or delay any flight, right of carriage or reservations (whether or not confirmed) and determine if any departure or landing should be made, without any liability on the part of UA. UA may re-accommodate Passengers on another available UA flight or on another carrier or combination of carriers, or may refund any unused portions of the Ticket in the form of a travel certificate.

Even if it was IDB, unless Costa Rica has an IDB compensation rule, no compensation is due per COC RULE 25 (B) anyway.

I would be looking for more than $400...

What immigration forms are we talking about here?

I've never been to Costa Rica, but it sounds like OP made it to the gate, which in any country I've been to means immigration formalities for that country are complete.

Was it the US customs declaration? It doesn't make sense to deny boarding for that not being available, the OP could have just picked one up in the ORD immigration hall. Indeed, I've been on flights where they ran out of forms, no big deal, just a minor inconvenience in having to fill it out in the immigration hall.

If it really was the US customs form, I'd push for serious compensation because the denied boarding was completely unnecessary and due to ground staff ignorance.

The concept of force majeure is applicable when there are circumstances beyond the control of the parties to a contract, which is not the case here if UA's improper handling of immigration forms is what resulted in the government intervention.

I went through Costa Rica a few years ago and they do have a nominal exit fee on all passengers flying out of the country; IIRC after paying this fee, the authorities put a stamp on something and this is later checked somewhere before security (or possibly even before the check-in counters?). My memory is hazy on this, so I can't remember when exactly they check the stamp. But you are right in that it's unlikely this would be the reason for denial of boarding, since they check the stamp long before the gate.

While employee stupidity is somewhat widespread these days, bungling documents is not yet a force majeure, I believe.

This was - according to UA itself - not an oversell situation. So even their own rules would not apply.

UA - based on no failure of the OP - simply has refused transportation and hence not honoured the contract.

this might be a more pertinent CoC rule

+1. Today in ORD after a missed connection due to UA not staffing crew from originating city causing my trip to become worthless (missed meeting) I told CS of what happened. Once I told her my status, she immediately wanted to ensure I was helped. It wasn't that she didn't want to help prior to disclosure, but once disclosure occurred, it was a fait accompli I'd get what I needed.

That's a good question. Perhaps the term IDB does has a strict definition pertaining only to oversales, which would mean that OP's situation doesn't technically qualify. However, this doesn't negate UA's fault and responsibility in this matter. Even if CR has no laws stipulating an appropriate level of compensation, IMO UA should compensate at a level comparable to an IDB situation stateside.

The traveler's documents were apparently in good condition when UA took possession and therefore responsibility for them. This rule doesn't cover the circumstances of UA negligently losing, or misplacing, the travel documents.

Did you sign anything when UA gave you the $200 voucher?

except to say I'm not suggesting UA is off the hook but IDB seems to be the wrong path prusue

OP presented the required documents just fine. That section is non-applicable.

If anything, what UA has done is worse than an IDB as OP wasn't offered comp voluntarily. But it wasn't outright fraud either as UA did then offer rebooking and hotel and a measly voucher.

Why not ask for something in the spirit of EU 261 - and file an IDB DOT complaint if denied?

Just for the sake of discussion(not argument!) the passenger didn't fail to obtain and present documents so the second part of the quote doesn't seem to apply. As to the first sentence, can losing documents be construed as "assistance?" I think that's a pretty weak, but not impossible, interpretation.

On what grounds since DOT lacks jurisdiction per the earlier quoted CoC (rule 25)?

I think filing a complaint with UA is the first step -- claiming IDB or DOT or any other nuclear option just distracts the discussion and provides UA an easy out.

The earlier question about a VDB form is important because if such a form was filled out, it is going to mess up the claim.

Agreed 100% on status helps.

I had similar situation. mid-December, I was flying on Lufthansa from FRA to EWR when we had a medical emergency and had to land in Dublin. We had a 3+ hour delay and almost everyone who had connections missed them, including me on the last flight to PBI. As a 1K, when I landed there was agent waiting for me with a boarding pass on the last flight to FLL that night. I noted there were probably 2-3 agents there with a total of maybe 5-8 boarding passes. Considering we landed around 7pm and the hundreds on board, it is safe to say those passes were for elites.

So, while I think UA service has gone down in many ways, I have to give them props for staying on top of my flight.