So, I found the answer. You need to replace the single-quote (') with double-quote (") and then escape the double-quote inside the JSON string (\"). I think this should be mentioned in the Acralyzer documentation so that Windows user don't need to go the extra step to find the solution.
curl -X PUT -u myUsername:myPassword http://kaydee.iriscouch.com/_users/org.couchdb.user:myNewUsername -d "{\"name\":\"myNewUsername\", \"password\":\"myNewUserPassword\", \"roles\":[\"reporter\", \"reader\"], \"type\":\"user\"}" -H "Content-Type: application/json"