When the packet is inside the bottle and you are not squeezing, the bubble is large enough that it will make the packet float very nicely. However, when you squeeze the bottle, you increase the pressure inside the bottle. This will compress the air bubble, which will increase the density of the packet. The packet will now sink. This is a classic experiment called the Cartesian diver, updated for a modern flair in the spirit of the Little Shop of Physics.
Submarines use a similar principle to control their buoyancy, as do some fish.
Other Things to Try With our packet, the buoyancy is also affected by temperature. We are a travelling program, so our experiments get left out in the van overnight frequently. When the bottle gets too cold, the packet will not float any more. Currently, we are working on making a Galilean thermometer by putting a bunch of different packets in a bottle, each of which has a specific temperature at which it makes a change between floating and sinking. The packet also floats or sinks depending on air pressure - which varies with altitude.