What kind of job was the director-level position?
Companies can require that applicants are British citizens, but it does usually depend on the type of job it is - because there may be specific reasons why the applicant must be British.
For example, government positions and jobs requiring certain security clearances (i.e. police, military, politicians etc.) usually require that applicants are UK citizens, because of the nature of the work.
For my job, working in a government organisation and handling restricted and up to top secret documents, I had to get full UK government security clearance and I also had to show that I had been living in the UK for the last 5 years. Ideally applicants needed to have UK citizenship, or at least ILR in the UK. They could still apply if they didn't, but they may have been restricted in terms of what information they would be allowed to handle in the job (i.e. they may not be allowed the same security clearance or have restricted access to certain departments in the company).