That is a VIPR operation which is completely different from CBP/BP/ICE operations at airports. VIPR operations are when the TSA (BDOs/FAMs) go out somewhere, usually to a train or bus station, with the stated goal of preventing and detrrring terrorism by being random and visible. But that would produce no tangible results since they are extremely unlikely to catch any terrorists, so they bring along the local law and some other federal alphabet soup (DEA/ICE/BP) depending on the operation. The operation then becomes about finding and arresting criminals and unlawfully present aliens with terrorism as as the excuse and BDOs end up looking for 'suspicious' brown people while drug dogs sniff away. In other words, the temptation of running a dragnet under the excuse of (and possibly with the funding for) preventing terrorism is unsurpassable.

Compare this with BP checkpoints set up in airports-- these are unrelated to VIPR. These can work one of two ways: The first way is a BP officer standing behind the TSA TDC looking at the documents presented to the TDC and stopping passengers whose documents indicate that they are potentially aliens and then verifying status-- another variation of this is a BP officer in the same location asking everyone if they are US citizens and verifying the status of those who aren't. The second way is to station a team of BP officers in the jetway of a flight and ask boarding passengers if they are US citizens and verifying the status of those who say they aren't; each way has its pros and cons.

The chances of encountering a BP officer at ORD doing an immigration checkpoint are extremely slim due to allocation of resources; not only is ORD a huge airport with numerous checkpoints and flights, but Chicago is not a common origin for air travel by unlawfully present aliens as compared with other airports. The same cannot be said of FLL. Airports near borders (including maritime borders) are good target airports since those airports yield a steady influx of unlawfully present aliens making the trip North to settle at their final destinations. SJU is probably the most popular airport for these types of checks; other popular ones include LAX, SAN, BUF, PHX, MIA and similar. It is possible for FLL to have this type of check though it is not something I've seen there.

Always a good idea.

Not always an option.

SecureFlight is part of the TSA; it does not run things through the USCIS/CBP database.

Were they checking IDs or just asking citizenship, or both?

Again, note the difference between VIPR and airport BP operations. (There are also plenty of stand-alone BP operations).

Making a false claim of citizenship is a big deal; having a valid DL, perfect English and white skin makes getting caught very unlikely. Nervousness would be the only thing to worry about-- and any BP officer worth his salt knows that IL is one of a few states that doesn't verify status when issuing DLs so additional investigation would be done.

This can be sorted out before or after an "unfortunate incident" in point of fact; sometimes it can only be sorted out after the fact as a practical matter. An alien who entered with inspection is at a signifigant advantage.