YouTube information on a user, would deleting the account delete their information on the person?

Deleting the account most likely will not remove all traces from YouTube's internal systems. The first thing to do, of course, is to have the person who posted the video remove it from YouTube if that hasn't been done already. Of course the damage may have already been done given that the company is apparently aware of it.

The kind of identifying information being sought by the employer (such as email address, IP address, etc.) is protected by a strict federal privacy law, the Electronic Communications Privacy Act of 1986. Most online service providers will therefore not divulge account information until they are served with "legal process" (i.e., a subpoena, court order or search warrant). To do this, the employer would need to file suit against the person suspected of posting the video (or "John Doe" if they don't know who posted it) and then issue a subpoena to Google (the owner of YouTube) to turn over the information. Google/YouTube should have information posted on its site regarding subpoenas/legal information requests, and it's worth reviewing their policies in detail.

You haven't indicated the nature of the video. If posting it involves criminal conduct -- as opposed to mere mischief which could get you fired -- I recommend retaining a criminal defense attorney who understands Internet-related issues.