You are not supposed to use grease in there, just oil. Grease will keep the pawls from working properly. There aren't any big loads on the freewheel components when the freewheel is freewheeling, so it doesn't take that much lube anyway.
However if the freewheel gets too gummed up then the little pawls will stick and the thing will freewheel both directions. (not good). I had a freewheel that did this. I was going to flush and lube it, but it had a stiff seal to keep stuff out. I used a small screwdriver blade to force open the seal slightly and used the thin straw on my spray lube to get some lube in there. It worked fine afterwards.