Hi, zeebra, I'm also interested in lab animal medicine (I'm starting a residency at MIT in July), and after having run the gauntlet of residency applications and being active in OSU's ASLAP chapter, I can probably give you some good advice!
Dr. Lesley Colby from Michigan has written a pretty comprehensive paper with some information that would answer a lot of your questions (it's a little technical/statistical/mathy, but still informative) - you can read it at
http://dels.nas.edu/ilar_n/ilarjournal/48_2/pdfs/v4802Colby.pdf. Some key points:
-Training program (ACLAM-recognized residency) breakdown by institution type - 60% public university, 22% private university, 6% government, 12% other.
-There's also a list of all the training programs, or you can find them at
http://www.aclam.org/education/training/location.html.
-Training programs are HIGHLY recommended for prospective ACLAM diplomates. You have access to mentors and co-trainees who can help you study. Additionally, the "experience route" to sitting for ACLAM boards requires 6 years of lab animal experience and a first author paper, and it can be tough to get in your research if you're working full time as a clinical veterinarian. You can still practice lab animal medicine if you're not boarded, but ACLAM diplomate status is a highly prized skill in the job market.
I can't speak on an MPH, but if you're looking to become any kind of administration/faculty, a research-based MS/PhD will definitely help you out.
If you want more information, you can look into joining
ASLAP (if you're a vet student) or
AALAS (open to all members of the lab animal community). ASLAP also has a list of learning opportunities (most are paid) for vet students at
http://www.aslap.org/career.phphttp://www.aslap.org/career.php.