LM2577 based boost converter board gives random shut-downs with a 3.7v Li-Po
How much current are you drawing from the LiPo cell?
It look like there is a protection circuit on top of the cell. (As there should be)
The protection circuit will disconnect the cell if case of over current or under voltage in order to protect the cell. 
I might just be that you are simply overloading the cell. That would explain the ~0.6V. It is probably the remaining charge left on the input capacitor of the booster circuit.