My advice is to keep social media and employment separate (unless it involves LinkedIn). It's a good practice especially as many companies frown upon employees using social medial for unofficial business purposes. By not asking your former supervisor for a reference on Facebook, you're not only keeping social media and employment separate, you're respecting your supervisor's right to do so. If you tried other modes of communication and he or she still hasn't responded and you want to reach out to him or her over Facebook, I recommend that you use the Instant Messaging feature to keep the communication private.
You should also contact the organization's HR department for information about the company's policy on references. If you have your employee handbook from the time you worked there, you should find the policy in the handbook. Many organizations restrict who and what information can be provided in a job reference. In the past, many former employees have sued former employers for unfavorable references even when the references were truthful and companies were held accountable. To protect their interests, many companies require that only employees in the HR department may provide references when requested in writing bearing the former employee's signature authorizing a release of information. Even in those cases, the information provided to a prospective employer may only include dates of employment and last position held. Salary information is usually provided only in cases involving financial transactions such as loan and mortgage applications.