Can I use pushbuttons with pull-updown driven by a gate?
The circuit as you have shown will work OK from a high level viewpoint. It will not provide clean single clock cycle transitions as the switch changes state. There are multiple reasons for this as follows:


The switch contacts will bounce and open and closed rapidly for a number of cycles each time the switch is pressed or depressed. This will allow the 1MHz signal to pass to the FF input in erratic spurts at the transition time.
The switch contacts will make or break with asynchronous timeing with respect to the 1MHz signal this will result in the FF output not outputting a clean divide by two of the 1Mhz signal. 
The same asynchronous behavior as above will result in some of the 1MHz signal getting through to the FF with very narrow pulses. At times these narrow pulses may cause the FF to go metastable where it may take a number of clock cycles for the FF to recover to normal behavior.