Te explico qué significan ambos términos, pero me temo que no te puedo dar una traducción ya que, al menos en España, no existe equivalente. A mí me ha salido en cuestionarios para médicos y al final el cliente ha tenido que suprimir la pregunta, puesto que es absurdo preguntar a un médico español "Are you board certified/eligible?" cuando no es algo que se pueda hacer en España. Todo dependerá, por supuesto, de a quién vaya dirigida tu traducción.

"Board certified" es el médico que, aparte de haber obtenido la especialidad
siguiendo los cauces normales (como en España) y que, por lo tanto, está
autorizado para ejercerla, ha completado un programa educativo voluntario aprobado por las llamadas "Specialty Boards" y un proceso de evaluación que
incluye un examen diseñado para evaluar los conocimientos y la experiencia
necesaria para proporcionar asistencia médica de calidad en esa
especialidad.
"Board eligible" es el médico que cumple los requisitos para presentarse al examen para ser "board certified", pero que no lo ha hecho. La American
Board of Medical Specialties, que incluye las 24 Specialty Boards, no
recomienda el uso de este término, ya que muchos médicos lo utilizan durante
toda su vida sin intención de presentarse al examen.
En resumen, son títulos que conceden unas entidades que supuestamente
garantizan unos mayores conocimientos del médico y una mayor calidad de su
atención.

The American Heritage® Dictionary of the English Language: Fourth Edition. 2000.
http://www.bartleby.com/61/48/B0354800.html

board-certified
ADJECTIVE: Having completed the process of board certification in a specialty field. Used chiefly of physicians: The hospital maintains a staff of board-certified surgeons and anesthesiologists.


board certification
NOUN: The process by which a person is tested and approved to practice in a specialty field, especially medicine, after successfully completing the requirements of a board of specialists in that field. For a physician, board certification is required in order to have the privilege of practicing in a hospital.


http://www.medterms.com/script/main/art.asp?ArticleKey=8725

Board certified: Board certified in medicine means a physician has taken and passed a medical specialty examination.

Board eligible, by contrast, means that a physician has completed the requirements for admission to a medical specialty board examination but has not taken and passed the examination.

For example, a physician must do 3 years of training in an approved pediatric residency to be board eligible and then successfully complete a comprehensive written examination to be certified by the American Board of Pediatrics.


http://www.abms.org/
What is the ABMS?

The American Board of Medical Specialties (ABMS) is an organization of 24 approved medical specialty boards. The intent of the certification of physicians is to provide assurance to the public that those certified by an ABMS Member Board have successfully completed an approved training program and an evaluation process assessing their ability to provide quality patient care in the specialty. The ABMS serves to coordinate the activities of its Member Boards and to provide information to others concerning issues involving specialization and certification of medical specialists.

The governing body of each Member Board is comprised of specialists qualified in the specialty represented by the board. The individual Member Boards evaluate physician candidates who voluntarily seek certification by a Member Board of the ABMS. To accomplish this function, the Member Boards determine whether candidates have received appropriate preparation in approved residency training programs in accordance with established educational standards, evaluate candidates with comprehensive examinations, and certify those candidates who have satisfied the board requirements. Physicians who are successful in achieving certification are called diplomates of the respective specialty board. The boards also offer recertification for qualified diplomates at intervals of seven to ten years.

Espero que te sirva