Macrophages can certainly phagocytose living cells, as you can read in Sompayrac's "How the Immune System Works", Chapter 1 and 2, but they do need an "eat me" signal of some kind in order to do it. For a bacterium that might be lipopolysaccharide and mannose. For endogenous cells, there is a balance of "eat me" and "don't eat me" signals. Apoptosis, in addition to neatly packaging up potentially toxic intracellular components, involves downregulating the don't eat me signals and upregulating the eat me signals. Macrophages can phagocytose live cells that seem to have some sort of problem (for example, tumor cells), but they are much more efficient at phagocytosing apoptotic cells.
As far as why the neutrophil has to die, in addition to making them more efficient targets for phagocytosis, they die pretty quickly anyway. There is some debate about the exact length of their life span, but from the time they're released by the bone marrow, the average circulating life span is probably less than 24 hours, regardless of whether they exit the blood stream as part of an inflammatory process or not. The exact function of this short lifespan is the source of some speculation, but we do know that, after being recruited in the inflammatory response, apoptotic neutrophils help suppress that inflammation, playing an important role in regulating the processes that sent them out into the tissue in the first place.
Sompayrac's book is a good overview, but if you want to read the gory details this is a good review.