That question covers a lot of territory. First off, a 10440 cell has over two times the voltage of an Eneloop AAA. If you are going to be using it in a light that clearly states that either is acceptable, no problem. If not, you run the risk of damaging the driver/LED, possibly to the point of failure.
Assuming a 10440 cell is acceptable, they will contain a significantly greater amount of energy that an NiMH rechargeable or primary non-rechargeable AAA cell. If that is not an issue, an NiMH cell has more recharging options and is somewhat safer to recharge than a LiIon cell like the 10440, especially if one is not conversant with all of the ins and outs of recharging LiIon cells.
As far as eco-friendly, both types of rechargeable cells are more eco-friendly that the use of AAA primary alkaline and lithium cells. Given the number or recharge cycles either type of cell are capable of, there really isn't anything to make one stand out over the other from an ecological point of view.