Best way to amplify 3.3V square wave at 1MHz to 10V square wave at 1MHz
The cheapest way would probably be using a CMOS hex inverter 40106 which can be operated at up to 15V and biasing the input near below the switching threshold using a voltage divider and either AC-coupling the input signal via capacitor or DC-coupling through another resistor.
The signal will get inverted. If this is a probem just feed it into another inverter (there is a total of 6 in the IC package).
Of course it can't provide the current you need. That could be accomplished by a discrete CMOS inverter stage (discrete P-MOSFET/N-MOSFET-pair). 
In addition if you use two drivers, one at each terminal of the inductor (= H-bridge configuration), you don't need a center tapped inductor to create pulses of both polarities and you don't need a negative supply rail. Depending on which terminal is high and which is low you can provide the appropiate current direction.
See circuit diagram below:







simulate this circuit &ndash; Schematic created using CircuitLab
The inverter should be a 40106 operated at 15V.

Alternatively:
There are also more specialized and still cheap level converter/driver ICs like the MC34152,
DS0026 or ICL7667 which could also fit for your application. I think they all can drive up to 1.5A.
In any case I recommend to build and test it first on a prototype board before manufacturing a PCB. That will be much more meaningful than any SPICE tests.