Voltage Threshold Gate Circuit from 0-5V Input
Is there a better, more efficient way to handle this?


There are brownout protection and power monitoring chips that solve this problem in a single chip. 

I am not sure whether there is one tuned to your specific requirements (5.4 V nominal voltage, TBD threshold voltage for turn on).


  Is there a risk that when the OpAmps are at a negative voltage output (on the labeled OpOut2), will that cause a problem in the circuit?


Since your op-amp doesn't have a negative supply voltage, it will never produce a negative output voltage.

One more note: You have Q1 configured as a voltage follower. The voltage delivered to your load will be somewhere below 4.8 V (depending on the specs of TL081, which I haven't looked up). If you want to deliver 5.4 V to the load, you should use a PMOS transistor in that location, and pull its gate low when you want to enable the load.