In some countries (e.g. Australia), if you are an independent contractor, you own the copyright of the code, for which you give a perpetual license to the people hiring you.
If you are an employee, the code produced by you is owned by them, it is usually very hard to show what you do on your "own" time. But this is the reason they often wnat to have you as an employee and not an external source.
I am not sure about the US, but I believe is the same. Probably to reinforce this is that most companies ask you to sign those lengthy contracts...
cheers
