Why Is It Squishy Around My Eye?

Squishy??? Your globe (eyeball) is in a hole without any bone around it. What you are feeling is muscle and tissue. This allows your globe to move easily within the orbital space provided. The noise you hear is probably the "sucking" sound when your lids get separated from the globe. The lids stick to your globe because tears are always on the globe adhering the lids to them so dust and other irritants don't get up there.
Although it won't harm your globes to rub them, try not to do it very hard so as not to irritate your eyes.