Push pull-circuits of that design are notorious for fusing through due to inadvertently turning on both mosfets simultaneously.
Obviously, this can happen during switching, but it can also happen as the power is applied to the circuit. The current pulse is normally very short, however, the smaller the mosfet devices the more probable a failure will occur on one or both of them.
As such, when using rail-rail push-pull drivers like this it is required that some protection be provided to ensure that the current can not spike through the bridge.
Below is an example that uses an in-line inductor as a current choke.
simulate this circuit – Schematic created using CircuitLab
L1 and D1 in the schematic above should be sized to limit the rise time of the current to be significantly less than the switching time of the mosfets.
Resistor R2 should be included to force the circuit into a particular state while the logic that is driving it is powering up. This is especially true if the signal originates from a micro that is initially configured as a high impedance pin. Whether this resistor is pulled to ground of logic 1 will depend on which state you want the output to start in.
C1 is intended to try and protect the mosfets from any start-up voltage spikes on the power supply.
R1 should also not be over-sized. It needs to drain the capacitance of M1 and charge M2 quickly enough when the transistor turns off.
Ultimately, with this type of driver, it is preferred that separate control signals be used with a built in dead-time where both switches are turned off before one is turned on. In addition to giving you more protection for your driver, it also adds the functionality of being able to disconnect the output entirely.