I live in New England as well. You can use conduit underground, just make sure the joints are sealed properly. You wont need much, a dedicated 15amp breaker is fine. You can run the pipe and wire yourself and then get someone to install it into the panel if your not comfortable with that. You can also run the conduit and simply plug the end into an existing outlet somewhere during the winter months.
There are several things to do for water in the winter. Heated dog dish, a warming mat under the water bowl, getting a large plastic waterer and inserting a very SMALL water heater for a fish tank into it,5 gallon size is perfect. Their waterproof and will be fine submerged.
I bought some nipples from Neil Grassbaum on this forum and installed them into the bottom of a 5 Gal pail. Come winter, ill just drop in a fish tank water heater and the problem is solved. It will only come on if the temp is low enough and it solves the problem of dirty water.
Make sure you run the conduit up inside the coop right to the electrical outlet so, they wont peck at the wires. Adding a GFCI outlet is a good idea as well but you can also just add a GFCI breaker at the panel and it will protect everything on that circuit.