While I assume nuclear power would be better suited for the amount of juice they need the new all electric Zumwalt class (including propulsion) along with integrated power system uses gas turbine generators and produces something like 10 times the electricity of other destroyers. So nuclear power isn't a requirement but I'd think it would be ideal, then again I don't know enough about naval nukes to even be dangerous.
I doubt many, if any, other existing ships would be able to realistically mount and use railguns though but I'd wager that anything being designed or produced right now will be.
The Zumwalt can generate 78 MW of electrical power. The 2 reactors aboard a new Ford class carrier can provide over 600 MW
Certainly turbines can do the job, my previous post was to highlight that at some point it may be more sensical to use a naval reactor to generate the necessary energy. You can keep installing diesel or turbine generation capacity plus the fuel storage necessary but you now have a bigger ship, the majority of it increasingly devoted to just power generation for this weapon. Nuclear power offers the ship unlimited energy for this power hungry weapon but also the important ancillary benefit of unlimited propulsion.
At one point the US Navy had a nuclear power cruiser and destroyer (15k and 9k ton vessels) so the reactor technology does exist to build small ships that can power a railgun
Keep in mind that these railguns, should they succeed and be more widely deployed are going to be considered first generation railguns. There will be better and more powerful railguns developed with increasing electrical power requirements.
I do not know the direction this new technology will take,what role the navy will utilize it nor what class of ships will be constructed to take advantage of this. But given how much speculation is out there about how this type of weapon might supplant carrier based power projection, one cannot rule out nuclear power as the most likely power source.