Can a company dictate what an employee can or cannot post on Facebook?

To a certain extent, yes they can, however, do note that the law in this area is constantly developing.. and fast. If you have a serious concern, do not hesitate to consult with an employment attorney regarding this. It, of course, is fact specific and may be subject to handbooks and employment contracts. I continually advise people to be very careful regarding what they post on Facebook regarding other people, their employers, employees.. really anyone. Best of luck.

The National Labor Relations Board ["NLRB"] has issued guidelines describing how it will enforce its [extraordinarily liberal-biased] interpretation of labor law as applied in the social media context. You can read a summary of that guidance and its various reports via the first link below. The reports describe examples of when disciplinary action is proper and when it's not.
The National Federation of Independant Businesses has created a useful checklist extracted from those reports [visit the other link below].
You should speak with your company's employment attorney to learn how your company has decided how it intends to comply with the relevant laws. The NLRB's interpretation of those laws are NOT the final say [the federal courts have the final say] and will undoubtedly change once the current administration is replaced by a more reasonable, conservative administration. But, for now, the NLRB has the authority to enforce labor law in its own, peculiar way.

The Company needs to be careful here. The National Labor Relations Board has recently come down on employers (including private employers with non-unionized workforces) that take adverse action against employees for their Facebook and other social media postings if those postings interfere with an employee's right to engage in "protected concerted activity." Generally, there may be protected concerted activity when two or more employees act together to improve their terms and conditions of employment. That could include engaging with/reaching out to other employees in social media spaces.
The clinic should hire a lawyer to review its policies and practices concerning social media to make sure that they do not violate the National Labor Relations Act. In the meantime, check out the NLRB's website for more information.

As long as the company does not violate the NLRB rules (which usually relate to employee to employee communication) they can institute any social networking policy they chose.
I have worked for many companies creating rules for their employee handbooks (e.g. email belongs to the company, clients belong to the company, etc.) These policies are legal and proper and defying them is cause for a disciplinary action.
You may want to discuss your situation with a lawyer in more detail. Most lawyers on Avvo offer a free phone consultation.