Re:You may still have a non-compete in place (Score 1)

One thing many have not mentioned that could provide a legal basis for the employer's threats is if you signed a Term Employment Contract--i.e., you contracted to provide your services for a set period of time or until the completion of some project. In such a circumstance, your resignation (even without moving to a competitor) could be a breach of your term employement contract, and your former employer can sue for lost profits you'd make. Think of it as a pro sports contract: Lebron James on your team means increased ticket sales, and his leaving while under contract will result in lost ticket sales that the team could recover through a lawsuit if he breached the contract. This circumstance is often overlooked in American labor law because very few people sign term employment contracts.

If you were an at-will employee, the people advice everyone has given should be correct.

As I haven't discerned the locale of this case... This is not legal advice upon which you should rely. You should find a lawyer in your local jurisdiction competent and experienced in employment litigation. (I know it's lame, but it's lawyers and the law.)