Unfortunately I won't solve the problem for you. Your question was interesting to me, but you hadn't shown any effort to solve the problem yourself. Which might be because you couldn't even begin.
I don't have enough karma to comment on the question directly. Which is why I am posting this as an answer.

1. For a short explanation of how the force relation works in STR, look at page 41 of this document.
2. Also as Eric suggests, the form of the spring force does not generalize uniquely to A relativistic force field, which is perhaps why it does not have an unique satisfying answer. You could try to classify the set of potentials which reduce to the spring force potential as $c>>1$. This would be why i was intrigued by the question. Do expect an edit from me in the future with the results of my feeble attempts at a solution to this.
3. Although the spring force might not be a good potential to look at to gain intuition in STR, it might be an interesting problem in its own right. However if you still want you can try to simulate two charged particles in the relativistic EM potential of each other. It would be rather fun to observe and you might test out the Lienard-Wiechert potential. Or, sigh, you can play this free game. My suggestion - do both.
4. And do post a link to your simulation here when it is done. Would love to see your work on it!

Your question was interesting to me, but you hadn't shown any effort to solve the problem yourself. Which might be because you couldn't even begin.
I don't have enough karma to comment on the question directly. Which is why I am posting this as an answer. Please don't vote on this at all, thus!

1. For a short explanation of how the force relation works in STR, look at page 41 of this document.
2. Also as Eric suggests, the form of the spring force does not generalize uniquely to A relativistic force field, which is perhaps why it does not have an unique satisfying answer. You could try to classify the set of potentials which reduce to the spring force potential as $c>>1$. This would be why i was intrigued by the question. Do expect an edit from me in the future with the results of my feeble attempts at a solution to this.
3. Although the spring force might not be a good potential to look at to gain intuition in STR, it might be an interesting problem in its own right. However if you still want you can try to simulate two charged particles in the relativistic EM potential of each other. It would be rather fun to observe and you might test out the Lienard-Wiechert potential. Or, sigh, you can play this free game. My suggestion - do both.
4. And do post a link to your simulation here when it is done. Would love to see your work on it!

EDIT1: This paper takes on solving a relativistic Harmonic Potential for those who are interested.