I'm anti-health regeneration. To be honest, grabbing a nice meaty healthpack that restored 50+ hp after a tough boss encounter or ambush feels very rewarding. An example of this, for me, is Max Payne. After a tough section of corridor shooting, you'd be low on ammo and health. Then you open the room, get your ammo stocked, four packs of pain pills, maybe a new gun. The pain pills felt like a reward for getting so far, having excess pain pills felt like even more of a bonus. I think players need healthpacks in games to work as an incentive. A carrot on a stick if you will, it's a small and necessary reward, but it rewards the player nonetheless.
I also feel health regen cheapens difficulty. You could theoretically wear a boss down by standing up, popping a few shots, sitting down, regen, stand up, pop a few, and so on. Rather than the actual skill of taking the enemy down with a set amount of health. Fighting games for instance would not be fun if your health would regen, you and your opponent would run around the arena until it was full again. You have set life, you have a task to beat the enemy. Shooters should follow this example.