My question involves labor and employment law for the state of: New york

I got an offer from a company and I signed employment agreement with them on Apr 21. Its a consulting company and job will be at their client site. I am on H1 visa and they filed my LCA and not yet started with H1 transfer. On May 6 I decided not to join them and informed them. But they are saying that since I am breaking the contract I have to pay for all the damages. And they are saying that it can be huge amount since it affect their relationship with the client.

This is what there in employment agreement in Termination section:

Nothing herein is intended to alter the at-will nature of the employment relationship. Both Employee and Employer may terminate this Agreement, without any reason by giving the other at least fourteen (14) days prior written notice of the termination. This Agreement may be terminated by Employer immediately if Employee fails to perform his duties or engages in any act of dishonesty or is charged with the commission of a crime that, in the sole judgment of Employer, affects Employee's ability to carry out the terms of this Agreement. Either party may also terminate this Agreement for a violation or breach of this Agreement. In the event of termination for breach, the non-breaching party will give the other party at least fifteen (15) days written notice of the breach and an additional fifteen (15) days to cure the breach. If the breach is not cured within that period of time, this Agreement may then be terminated. Termination for breach will not alter or affect any other rights or remedies that may be available to the non-breaching party.