200A service is at 240V. That's 400A of 120V so you should be OK. Hair dryers, curling irons, etc. at each station will be on 120V. You should have a separate 20A circuit for each of the 14 stations. That could be with 15A duplex receptacles (allowable for 20A circuits), though 20A duplex receptacles will give you the most flexibility.
If every station is running at the same time (which probably won't happen as often as you might expect) that would be 16A * 14 = 224A, leaving 176A for everything else. "Everything else" includes small loads:
- lighting - which doesn't take much (relatively speaking) these days thanks to LED technology
- computer for the front desk
- TVs to keep the waiting guests entertained
and potentially 2 large loads:
- HVAC - primarily AC, as heating would (I hope) use natural gas. Air conditioning, particularly for a space generating 90,000 BTU/hr drying people's hair, can be a significant load. If you are in a big building providing HVAC via a central heating system and a chiller then there is no problem. But if you are in a space where you provide your own HVAC then you need to get a real-world estimate of air conditioning capacity and associated electricity usage based on not just the usual parameters (square footage, number of people) but including the hair dryers running a large percentage of the time.
- Water heating. My guess is you will use a significant amount of hot water at times for washing hair. If your water is heated with natural gas, no problem at all. But if you plan to use an electric water heater (or multiple electric water heaters) then you need to factor that in as well.
HVAC & electric water heaters are 240V loads. A typical residential water heater might have 9,000W (2 x 4,500W) elements and use a 40A circuit. Your requirements may be significantly higher as you have to allow for 14 shampoos within 30 minutes of the shop opening each morning.