Does employee noncompete contract apply to independent contractor?

In most states, I would not be able to answer this question, because the enforceability of a non-compete agreement turns on very fact-specific questions, and every situation is different.  California, however, has very strict laws against non-compete agreements (with some exceptions, such as where trade secrets are involved).  You should look at California Business and Professionals Code Sections 16600, 16601, and 16602.  There are even California cases which have held that it is an unfair trade practice for a company to try to enforce a non-compete agreement (known as a restrictive covenant) against a California employee.  Unless there are circumstances of which I am not aware, I don't think that the California courts would make you stop working in your field for 2 years.  As for "removing all work", I'm not sure what you mean.  If you mean to stop displaying work you have done for other clients on your website or promotional materials, I do not think that they can prevent you from doing this; it might be a different story if it was work you had done for them.