Impedance matching for push-pull RF amplifier
Going to a transmission line transformer will help extend the bandwidth, but probably not to 600MHz! 

The issue is that for a transmission line transformer (Or balun) to work well at high frequency the lines must be electrically short at that frequency, while for the thing to work well at low frequency you need the ferrite to add magnetising inductance, achieving both is hard (Above maybe 100MHz you will find that a transmission line transformer works just fine without any ferrite at all).  

Further, above maybe 60MHz or so the device parasitics start to become a real pain, and narrowband matching that can tune these out becomes somewhat desirable, you can build a broadband VHF/UHF amp, but it is always going to be a compromise, and a HF/VHF/UHF amp from a single device is really stretching it. 

Consider for example that the gate impedance generally falls with frequency (at least until the leadout inductance becomes dominant), at HF it is high enough that a swamped gate design works, but eventually the gate impedance becomes annoyingly low, and a narrowband match with an L network becomes the way to play (You could transformer your way down to very low Z and do a swamped gate design but gain would be very low because all the drive would be winding up in your swamping resistors without developing much voltage at the actual gate.  Similar considerations apply to the drain match.