Can I copyright my design of someone else's company/name?

If you design a copyrightable logo and the agreement is silent as to transfer of copyright ownership, you have the right to register the copyright and control its reproduction. Absent a written agreement to the contrary, the author (source of the creativity) is the owner of the copyright. Without looking at the logo, it is hard to answer whether the logo has the modicum of creativity necessary to obtain copyright protection. If the copyright office does not believe it is creative enough to be protectable, then you will not have any rights to prevent their reproduction of a logo. In practice, it is always better to put the terms of use in a written agreement before the logo is prepared. Without a written agreement, you are creating a situation that could result in litigation, hurt feelings, unrest, and frustration.