I am wondering, do blood cells die right after they leave the body?
No. If they did, blood transfusions wouldn't be possible (or they would require some sort of direct body-to-body system).
Red cells collected can be stored under refrigeration for over a month. See for example the description of what happens to donated blood from the Red Cross. Platelets don't last quite as long, but they also survive for some time in the right conditions. Of course, some cells die during storage, but there is no instantaneous die-off once cells leave the body.
Blood cells do not die immediately because they still have some amount of nutrients in them that they can use until it finishes. They only die when they lose their source of nutrients after using up all that is in them.
Cells in blood still have access to nutrients due the presence of plasma which helps keep them alive for while.
Put in a fridge(4-18°C), the temperature slows their rate of metabolism allows which further extends their lifespan. But can die of heat shock if frozen and thawed. You can read more on it in;