Why are cheese curds squeaky?
This article says it's not air and water: the squeakiness is caused by rubbing against the network of protein strands in the curds, and the squeakiness fades over time because the protein network deteriorates.

With respect to microwaving, it says:


  The reason microwaving helps reinvigorate some of the squeakiness is that the heat starts to alter the amount of moisture trapped inside the curds and creates another bout of hydrolysis, dropping more negatively charged hydrogen ions from casein proteins while they bind to the internal heating water.


I'm inclined to believe this because it explains why cheese curds are different from, say, cheese, and it's a more detailed explanation consistent with the article cited in Laura's answer (saying the protein networks are more elastic when fresh), which also explains the effect of the microwave!