Can company give IP rights away for application I wrote off hours to a startup?

For the company to own the IP, it would have to be a work for hire as defined in 17 USC 101. That requires that it be made by an employee withing the scope of his employment.

The author was an unpaid intern. Interns that are employees MUST be paid at least minimum wage under the Fair Labor Standards Act, so we may infer that the relationship in this case is not an employee relationship. Without an enployee relationship, no work for hire. (Specially commissioned works can be work for hire without an employee relationship, but only for specific kinds of works which are not applicable).

The request is even more questionable in this case because it doesn't sound like it is the COMPANY asking for the code. A director wants to give the code to his brother who has a start up in the same industry.

Personally, I'd report the company to the Department of Labor for possibly illegal internships, and I'd report that director to the other directors or owners who might not be aware that he's trying to use code their intern developed to help his brother's startup.

That's what I did when I was in your situation. This way, 1) You don't burn any bridges, 2) You're recognized for your work and 3) If you get other contributors to your project, your software will end up coming out on top.