You can add a verb to Twitch.api to specify the PUT or DELETE:
Twitch.api({method: '/users/:user/follows/channels/:target', verb: 'PUT' }, function([...]) {
[...]
});
Alternatively if you’d like to make the calls manually, the API supports a _method query string parameter:
https://api.twitch.tv/kraken/users/test_user1/follows/channels/test_channel?_method=PUT