Also, when and what training? SEALs, and all special forces, are constantly training, both at Navy and other military bases as well as at private facilities. VietNam era versus present will change somewhat where they are trained. What their specialty is will change training. Whether or not they are/were in other special forces groups can change training. Some might even skip Great Lakes, others may successfully go through BUDS and never become a SEAL.
In general, most SEAL trainees will be in their early 20's, though a lot has changed in the last decade as dependency on special forces troops has increased. Some won't even be Navy, but now days most will have joined the Navy, or Marines, with an intent of becoming a SEAL. Much more elan and cache now than the group used to have, though every branch of the service has a vast array of special forces troops now.
I have a friend whose son went through the Naval Academy and eventually tried out for SEALs (Dad's still pissed, SEALs don't end up on the Joint Chiefs, where he wanted his son). Made it on his second attempt, at age 27. Another friend's son joined the Coast Guard out of high school to become a rescue swimmer, eventually moved to SEALs and ended up in three BUDS classes before passing (injuries both time he was sent back). But another friend's son made SEAL the day after his 19th birthday (Dad was a VietNam era SEAL and the boy had trained for it since he was about three).
The guys my age that I know used to be SEALs were all sailors somewhere before choosing special forces, but in that era there wasn't anything for sailors to do so the only way to get to combat was special forces. That's a huge difference from the last 15 years or so. Same with Rangers and other special forces. Have a friend who left Marine Recon six months before 9/11. Tried to reup and they didn't want him.
So you have a wide range of possibilities for your character, Foreign-born SEAL candidates are not unusual and, if your character has special training the SEALs need, it's quite possible he'd be accepted. If your character is special enough, 72 hour liberty could be granted, or if he had cause that warranted it. Rare doesn't mean impossible. But I wouldn't see liberty during BUDS as a possibility without getting the privilege of returning later with a different class.
You might consider not making your character a SEAL. SEAL missions can, and do, include operators from other assignments, even foreign military, so you could write the character into a SEAL mission without making him a SEAL. I have a friend who was Army Rangers via 82nd Airborne, worked with the DEA on some missions, ended up in the CIA, went through BUDS (at 32) and did CIA ops until he retired (Yes, you can retire from The Company). He worked with SEAL teams in Central and South America quite a bit.
The sky, as well as land and water, is the limit.
Jeff