The answer is yes but it depends on the type of wire and insulation.
Individual wires labeled THWH are rated for continuous current of 65 amps, and THHN wire rated to 75 amps. As mentioned earlier, both of these options would be safe but must be run through conduit.
The NM-B wire mentioned above is commonly referred to as Romex, which is a brand name. This has 3 or 4 wires wrapped in a single sleeve, and is the typical wire found in your house. This will NOT work on a 60 amp circuit because #6 copper NM-B is only rated to 55 amps.
The wire has to meet the breaker rating even though electrical code limits the max continuous current to 80% of the breaker rating. So for your 60 amp circuit you must set the wall unit to 48 amps or less (earlier generation wall units could go up to 72 amps).