I had a couple of initial ideas, neither of which is perfect yet, but they may give an indication of the right direction to go:
I have added some animations to the blend file below:
spring-3.blend (1.23 MB)
Just press Play to see them and feel free to take the blend file apart so you know what I did. Of course you can ask any questions on what and why - for the armature solution, you need to pay particular attention to the bone IK properties. For the other, I think there are issues with using a curve modifier as this is tending to distort the spring profile.
I really need to see the spring you are trying to animate, either an image or blend file. :yes: