Can a current student/recent graduate get an H1B visa?
Yes, but it's not easy. You'll need to graduate, get an ECFMG certificate (after your graduation date), apply to take the Step 3 in a state that allows it before PGY-1 (e.g. Conneticut, Texas, Cali, Florida, Maryland, etc; NOT NY), take/pass step 3, get your score, THEN match, and finally apply for the H1B on April 1 (earliest you can apply).
This is impossible if you plan to start residency less than 7 months after graduation (e.g. if you want to start July 2013, the latest you can graduate is Dec 2012). Even then, it's pushing it, and you'd have to get an H1B-issuing residency program to take a big risk in taking you, while you scramble to get everything done for the H1B before April 1st.
Bottom line: if you're a student dead-set on H1B, expect to take an year off after graduation.
Whats the deal with the J1?
It sucks. It's sponsored by the ECFMG (if you completed all their credentials), and requirements vary based on country of origin (MCCEE and Letter of Need for Canadians). You have 7 years from the date of first entry on this visa to finish-up, and get (the f#$%) out for 2 years (the "home residence requirement" = HRR). You can't take any positions that go past pgy-7; and if you get into PGY-7, be careful your Visa doesn't expire a few weeks or days before you graduate.
If you enter on the J1, and you do a prelim/transitional year, and do not match into anything in year-2, you're pooh pooh-out-of-luck -- your 2 year HRR ban kicks in. (Once you enter the US on a J1, the ONLY way out of the 2 year HRR is an HRR-waiver job, or a few obscure exceptions, one of which is below).
*It is possible to do a second prelim/intern year if you have no other choice.
Also, on an J1 visa, you can only switch specialties ONCE during the first and second year of training, or before the end of the second year - once you're in the third year of training (residency or direct fellowship), you can't change specialties. You also can't Moonlight outside your residency hospital while on a J1 (you can on H1B, if you get the job, and insurance to cover you).
You can get a 3-year HRR-waiver job after (I don't know the prevelance of these, but there are less in surgery, EM, and any other (sub)specialties... they are mostly in non-specialty primary care/psych); e.g. Utah refuses any non-primary care HRR waivers (only allowed in FM, Gyn, IM, OB, Psych).
State-wise Policies on J1 Waivers : List of Links: Visa Basics for IMGs, FMGs, Medical Students, Residents, Physicians & Doctors
*If you get married while on a J1, you must STILL leave for 2 years before your are eligible for a greencard. In this case, you can try to get an HRR-waiver after residency if you can prove your spouse will face significant hardships without you (i.e. they don't make enough money--i.e. ask them to quit their job for a year). Or, like before, get an HRR-waiver job.
Doing a fellowship in your home country is a good way to finish your advanced training (if you have enough PGY years to meet your country's requirements), and take care of the HRR. If you want to do a fellowship in the US, your best bet might be to finish residency (3-4 years), get your HRR-waiver job right away (only in public hospitals; private clinics/hospitals are NOT eligible/NOT considered under-served), work 3 years, convert to a green-card, then go back to a fellowship.
If you still want to do a surgical specialty or other sub-specialty or get your medicine fellowship out of the way first, there may be less HRR-waiver jobs available for you, (but this is not confirmed, just hear-say), but you might find one; or you can go back to your home country and practice there (but for Canadians, that will likely mean ~6 months-1 year of licensing + VERY hard tests + 3 years contract in underserved area).
H1b vs J1 visaH1B versus J1 Visa Residency: Visa Basics for IMGs, FMGs, Medical Students, Residents, Physicians & Doctors
*Note the 6/7 year limits.
Only down sides of H1B visa are: 6 year limit instead of 7 (though you can try to convert to green-card after the 3rd year; and Fellowships are government subsidized for citizens, permannt residents, and J1 visas, NOT for H1B's, so you may have trouble getting a fellowship with an H1 if you're applying 2 or 3 years into IM, Peds, or Psych residencies, before getting a green card.
Are there any other visa options?
Sort-of. The K1 or K3 (engaged to a US citizen) will take about 9-10 months to get, plus another 3-4 months for a "Employment Authorization Document" application. Advantages: No 6 (H1B) or 7 (J1) year limit, can be converted to 2-year green card right away. Disadvantages: takes >1 year, lot of paperwork/hassle, have to find a citizen to marry. I couldn't actually find anyone who used this path, so consult a lawyer before you try this.
Or, if you have a million dollars sitting around, legally buy a green card: EB-5 Category : A Green Card that you can "Buy": Visa Basics for IMGs, FMGs, Medical Students, Residents, Physicians & Doctors