In this video he sudddenly regenerates a huge amount or health in about a second. This is because whenever a gunzerker uses his action skill (Gunzerking) he instantly regenerates nearly all of his health. After that the regen is still very good, but nothing like that first second. That sudden short burst of just crazy regen is gunzerker specific.
If you are looking to make another class regen like a gunzerker; good luck. You can make any class regen some but the amount of regen on other classes will typically be less.
The soldier has some minor regen skills like able (regen a small % of hp per sec for 3 seconds after damaging anb enemy), and Preparation (more total shield capacity + same regen as able but only as long as your shields are full), and a class mod that adds max health and a number of health regen per second (Note: this number is the direct hp per second, NOT a percentage of total hp).
Zero has an ability where he has hp regen while using his action skill, another skill (i think it's called resurgance) where you get some percentage of your total hp (4% i think but cannot swear to it) restored for killing an enemy, and a class mod with simmilar hp regen to the soldier one.
The siren has a phaselock related regen (elated i think it's called), a team hp regen class mod, the Sustenance skill (0.4% of your total hp per sec per lv max at 2% before class mod bonus'), the Life Tap kill skill (nice nod of the head to Everquest there), and the hp orbs from the Sweet Release skill (also linked to phaselock) can help quite a bit too.
The mechromancer has the skill Cooking Up Trouble that when maxed gives 2% health regen when clip is full, and Discord in the red tree that also adds 3% hp regen, so that'll get ya to 5%, and the With Claws skill also restores some hp. That said; regen is meaningless if you build the Mechromancer a certain way. Skip to 23 minutes in and watch the result of this guys build:
Bottom line, no one regens like a Gunzerker regens; HOWEVER, just because they regen crazy fast does not mean they are better than other classes. I personally hate playing a gunzerker.