For repeat-ability I would suggest that you require some form of PID temperature control. You could use a simple thermostat but that would not present a repeatable temperature profile given atmospheric variances, etc. This can be achieved with an Arduino as you mentioned - google "Brett Beauregard" for some great PID tutorials on Arduino.
Depending upon how tight the temperature control needs to be, and this does not sound like an application where you require it to be held <1C, I would suggest a simple thermocouple or RTD PID temperature controller with relay controlling the DC link voltage (see ebay for cheap ready-made PID options), that covers the temperature range you are after. This should allow you to keep the heating and cooling within around 10C.
Ultimately you can vary the DC link voltage to get the required power, but in the ZVS mentioned, minimum voltage is 12V, so nowhere near zero power output is possible without modification of the ZVS. Also, it would require a more complex setup (with modifications to the ZVS) to be able to continuously vary the DC link voltage, but for the purpose you have outlined I would suggest it is likely overkill (albeit more elegant and better controlled).
The issue with the switching approach however, is that the switching will cause additional stress and heat in your MOSFETS. However if you set the operation such that the relay operates at a maximum of >1Hz, you should be OK. The reason that the workpiece should not normally be in the coil at startup is because it causes the inductance of the coil to shift, which may inhibit oscillation. If self-oscillation does not occur, the mosfets will short out and burn.
Switching the DC supply causes large currents at startup when the workpiece is cold, and this is often less of an issue when the workpiece is hot (in the case of ferric materials). I have discovered in the past that the 'requirement' of having an empty coil at startup is not necessarily true and very much depends upon the relationship of the coil and the workpiece, and such operation IS often possible (albeit less sophisticated than it could be).
Since this is a DC supply, I am not sure what kind of SSR you will be using or where you hope to place it. A mechanical relay is possibly a better bet, placed between the DC power supply and the ZVS. I say this because if you switch the power supply instead, it will take time to come up to voltage, which is likely to cause problems with oscillation (many PSUs this is even a feature, "soft start"). By switching the DC Link with a relay it will be true on/off operation.
In short you will need to experiment, since your work coil, the size of your work piece, and its coupling to the coil will determine what you can achieve. I would suggest trying the relay option first as in many cases it works just fine with an appropriate interval (>1Hz). I would then reduce the DC link voltage manually as far as possible to provide:
- The required ramp time (heat-up)
- The minimum current draw required
- To reduce the quantity of on/off switching cycles demanded by the PID control
If switching proves unreliable than you will need to consider varying the DC link, which will require a power supply that may be controlled from the PID temperature controller. I would also suggest that you should include some kind of oscillation detection, and fusing to the device in the longer term as you could have issues with exploding mosfets and / or fire if left unattended and the oscillations failed, presenting a dead short to the power supply.