I planted 7 11/2 ft janed gold arborvitae a couple of weeks ago. They are planted in a 2-1-1 mix of top soil, peat, and compost. Surrounding soil is clay. I was watering every other day, about 1 gallon of water to each bush. I am seeing inside browning which I know can happen during a growth period and there has been some new growth on the top of each bush. My question is, how often should I water, and roughly how much should each plant get?

Your question is one that does not have a definitive answer. In your case, be sure not to overwater creating a overly wet situation given the rich amount of organic matter contrasting with the heavy clay soil surrounding the planting pit. Here in Illinois, we suggest checking every 5-7 days and watering only when needed. variables include air temperature, amount of natural rainfall, drying winds, etc. the only roots you plants have right now are in the immediate root ball. Surrounding soil may be moist, but unavailable right now. Water well before cold weather as your plants are alive year round.