An employer may ask employees to carpool, share hotel rooms, or do other things which are not illegal or discriminatory in nature while employees are doing company work. An employer may not force an employee to put someone in their car if they do not want to (and may be liable for safety reasons if someone is a danger), however, an employer may also let an employee go (unless there is a contract or employment or a written agreement that specifies otherwise) for failing to follow company policy.
When employees travel on company business, even if they use their own vehicle, the employer (as well as the driver) may both be held liable for any accident or injury that may occur during the "scope of employment" (unless the employee can be found to have taken time out for personal reasons while supposedly doing company business). This article and this one here discuss that issue. This may be an issue to discuss with a personal insurer as well as the employer, if that is something an employee feels comfortable doing.
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