From a design (and obviously ignoring realism) standpoint, the best system IMO for the single player experience would be something like: (The "units" I am using are arbitrary)
Regenerating shield type health of 100 units, that I'll just call "shields" from now on. Starts Regenerating rapidly when you are not hit for several seconds. This is your effective health for quick, strategic gunplay and such before getting behind cover again.
Old fashioned, traditional health of 200 units, with very minor regeneration. Regenerates 1 unit every couple seconds. This is your backup health that serves as a cushion in case you want to go further. Ideally, you don't want to have to rely on it.
"medkits" or whatever that heals the traditional health are very rare, pretty much all just appearing before a boss area so that the player doesn't have to wait to go into the boss at full power. Other kinds of powerups (including those that may temporarily enhance the shield) are more common.
For multi-player, there are three health modes that can be selected regardless of whatever other game-mode options you choose. The first just replaces all the health with only shield-type health (so just 300 units of shields). Medkits would of course not appear. This would likely be the most popular overall. The second would be all traditional health. Medkits are more common, and health wouldn't regenerate slowly like in SP. This mode effectively would be if you wanted to have a traditional-style FPS match. The last mode would basically be 150 shield, 150 health, just in case some people would like a mixture of the two (allows for using cover, but also means you can't just go all the way back to full after beating someone if they got through your shields, making you a bit more vulnerable). Medkits would spawn in this mode but would be rare, and the health doesn't slowly regenerate like in SP.
Weapon-wise, if single player has enemies with shields, then we can have some weapons that do better against shields and some better against other stuff. For multi-player, both shields and health would be impacted the same way for balance purposes. Rather than having something like the plasma pistol's overcharge for instant shield busting, have weapons with other effects instead. Like a weapon with a charge ability that disables the target's ability to shoot for a couple seconds, or which slows them down for a couple seconds, or which similarly is "compatible" with all three health modes.
Being able to carry 3 weapons is a must. You can put a 2-weapon option in MP, but being able to carry 3 of them in the SP campaign would be ideal, if you must limit weapon choices (you can also go the traditional route and just have no weapon limits in singleplayer). If you set it to 3 weapons, put a smaller icon of the "next" weapon next to the icon of the current one on the HUD so the player can get used to their current order so they know whether to tap weapon switch once or twice. And as long as we are on the topic of weapons...
If the setting can allow for it, remember to put some weird ones in. Perhaps a nice, unbalanced BFG equivalent too. You can simply have them not spawn in multiplayer if they are too unbalanced, outside of special game modes that can grant one to a team or something if they perform some task (every X kills, or something else). You can also potentially just make the multiplayer version(s) of these super-guns just single-shot-only.
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