What would have happened if the jet engine on my aircraft had ingested caulk from the pavement?
Well, not an expert on caulking, but birds go through plane engines sometimes, and they seem to be roughly the same consistency overall (if you account for the dirt and pebbles that are sometimes stuck to caulking.) So it possible they would have the same effect.  

If you're curious about what would happen if it's like a birdstrike you can take a look at the Miracle on the Hudson, or this 757 that had a birdstrike on takeoff.  With the 757, they simply went around and landed the plane since they only lost one engine...

Anyway, you folks were still on the ground, so all that would have happened is that the engine would have flamed out while you were sitting there and the plane would need to be taken to service to check for damage.  It would have been annoying to have to switch planes, but it wouldn't have been dangerous to any of the passengers or crew.

Nice of them to give you a free coffee though, considering you may have saved them a few hundred thousand in engine repairs if the next jet in that slot had managed to actually ingest the caulking ;).