What causes the squeakiness, and why does it disappear so quickly? Why does microwaving bring back a bit of squeak?
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!
According to Wikipedia, cheese curds squeak "due to air trapped inside the porous material".
Louisa Kamps wrote an article for the New York Times Magazine in 2004, describing it like this:
They squeal, most volubly within a day of their making, we learned, because their binding proteins are still superelastic, like new rubber bands.
Neatorama has an easier-to-understand explanation:
When curds are separated from whey, they are generally pressed, dehydrated, molded, and aged into cheese. If these steps are skipped, the curds last only a few days. These curds squeak because of the water that has not been pressed out.
The squeak is caused by the water and air trapped inside the curds. The water disappears as the curds become less fresh (or, in the case of most cheeses, gets processed into the smoother and or harder wheels or bricks we buy from the store). I'm not really sure of the physics behind why microwaving them makes them a little squeakier; I think it's just because you're drawing out the remaining moisture again.
Much of the cheese curds sold are refrigerated. But refrigerated cheese curds will never squeak. Leave the cheese curds at room temperature and if they are not wet, then add a bit of water to them to make them moist. If they are wet and at room temperature - NOT refrigerated - then they will most likely squeak. All the squeaky cheese I've purchased over the years has been at room temp. Refrigeration and dryness both turn off the squeak.