It is possible to remove with hot air without damaging the plastic, but it takes some skill and practice to do it.
The plastic can take these temperatures since it had to take them when it was originally soldered down. The problem when desoldering is that the large cold board is taking away the heat, so you want to make sure you heat up the board rather than the connector. Having a board preheater placed under the PCB helps a lot here.
Other components wont fall off because of solder surface tension, it can happen only to very large heavy parts like inductors or large transistors like D2PAK.
I believe ChipQuik is basically a low-melting bismuth-tin or bismuth-lead-tin alloy. There are lots of such alloys. "Woods" metal is one that has attained a little fame, but is not used in electronics. It is very low melting.
On point, I saw some devices the other day (BGA packages) that came pre-balled with low-melting solder balls. There is a chance that the solder beneath the connector is low melting. In other words, a low-melting solder paste was used for reflow soldering the device. I would give heating from below a try as others have suggested. There seem to be few other options.
If you heat from above, aluminum foil is a great protector.
I'm guessing ChipQuik is mostly Bismuth since it will form low temperature alloys when mixed with both lead or tin(lead free solder). The melting point of pure bismuth is similar to that of lead free solder and you do need to melt the existing solder in order to get it to alloy quickly and fully with the bismuth.
I have used QuickQuik low temp solder before and it does work well. It comes in useful when you want to desolder large chips of connectors where enough hot air to heat it all up might damage something. In that case i apply ChipQick to the pins and then use hot air to heat everything up at the same time and pull the part off. I don't use it everywhere but it comes useful in some more tricky situations.
Also clean up the ChipQuik really well afterwards by tinning and wicking the pads again. These bismuth solders are not as strong and could have undesirable brittle properties when they are in a random mixture of both solders with the wrong proportions. So cleaning is important.