In Connecticut specifically, if an employer hires a company to take photographs of their employees (headshots with nothing in the background that may be considered company property), do the employees have any rights to the photographs? (e.g. for personal use).
I don't know about Connecticut specifically but in most of the world the copyrights belong only to the creator of the work (the photographer) - unless there is a contract that say otherwise (work for hire, transfer of rights, etc).
The subject of the photo has no rights to use the photo what so ever - except for those specifically given by the copyright owner (or anyone who was given permission to give such rights by the copyright holder).
note that, depending on the situation the photo was taken in and the local law it's possible the copyright holder can't use the image either without permission from the subject - but this doesn't seem to apply in this situation
So, the employee has no right to use the photo unless given permission, probably by the employer (assuming the contract between the employer and photography company includes either a copyright transfer or a license to do whatever they want with the images)
The subject has no copyright rights, but they do have model (or "personal image") rights in certain uses of the photo.
You might have some say in whether that photo could be used in advertising or marketing type content, but otherwise, what Nir said is correct. A lot also depends on what your employment agreement with the company says, so each case like this is individual to a person and company.