I'd say you can use this analogy more for entropy, but you can say something about both.
The idea is that when you stretch a rubber band, you're actually forcing the molecules that make up the band to become less randomly distributed.
At full stretch, all the molecules are aligned with each other; this means that the system went from more disorder to less disorder as a result of the stretching.
When randomness and disorder decrease, entropy is said to decrease as well.
A stretched rubber band is an example of decreased entropy.
On the other hand, stretched rubber bands actually feel a little warm to the touch. This lets you know that heat is being released from the rubber band as its being stretched.
Thus, from an enthalpy point of view, a stretched rubber band is an example of decreased enthalpy, since energy is being released in the form of heat.
Bottom line, a stretched rubber band has less entropy than a contracted rubber band, and stretching a rubber band releases energy - exothermic process.
By comparison, letting a rubber band contract, which takes in energy - endothermic process, will lead to an increase in entropy.