The half life of proteins is mainly dependent on how quickly they are broken down by enzymes. Presumably when red blood cells are formed they lose these enzymes about the same time they lose their nucleous and (I think) their mitochondria.
The continual creation and break down of proteins allows their levels to be controlled by small changes to the balance between creation and degredation. It also helps remove damaged proteins.
If red blood cells do lose their mitachondria then the rate of protein damage should be much lower than normal cells, as the mitochondria leak free radicals which are the majour cause of damage.