Yes, they can file for a cap-exempt H visa ... if the job requires a BA and if the 'student' has a BA.
BUT, the 'student' will need to go home to apply for the visa.
There are special 'overstay' rules that may help the student.
PROFESSOR OF IMMIGRATION LAW for over 10 years -- This blog posting is offered for informational purposes only. It does not constitute an attorney-client relationship. Also, keep in mind that this is an INTERNET BLOG. You should not rely on anything you read here to make decisions which impact on your life. Meet with an attorney, via Skype, or in person, to obtain competent personal and professional guidance.
you cannot change your status here in the US and will be required to travel overseas and "consular process" if USCIS approves your H-1B to clear any previous out of status/unlawful presence here in the US. Depending on how long you have been out of status will make it more difficult if not impossible to obtain a visa overseas at the consulate/embassy overseas for the new H-1B visa since by departing the US you may trigger a 3 or 10 year bar on returning to the US due to your previous out of status condition in the US. You need to seek advice of competent immigration counsel prior to doing anything. Good luck.
The student has two options on these facts: return home and try to build his or her life there; get better education, find a job, build a family and so on; or stay here illegally hoping for immigration reform which may never happen, wasting his or her youth doing so.
NYC EXPERIENCED IMMIGRATION ATTORNEYS www.myattorneyusa.com; email: email@example.com; Phone: (866) 456-8654; Fax: 212-964-0440; Cell: 212-202-0325. The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter.