This can be done by building a custom class derived from click.Option, and in that class over riding the click.Option.handle_parse_result() method like:
Custom Class:
import click
class NotRequiredIf(click.Option):
    def __init__(self, *args, **kwargs):
        self.not_required_if = kwargs.pop('not_required_if')
        assert self.not_required_if, "'not_required_if' parameter required"
        kwargs['help'] = (kwargs.get('help', '') +
            ' NOTE: This argument is mutually exclusive with %s' %
            self.not_required_if
        ).strip()
        super(NotRequiredIf, self).__init__(*args, **kwargs)
    def handle_parse_result(self, ctx, opts, args):
        we_are_present = self.name in opts
        other_present = self.not_required_if in opts
        if other_present:
            if we_are_present:
                raise click.UsageError(
                    "Illegal usage: `%s` is mutually exclusive with `%s`" % (
                        self.name, self.not_required_if))
            else:
                self.prompt = None
        return super(NotRequiredIf, self).handle_parse_result(
            ctx, opts, args)
Using Custom Class:
To use the custom class, pass the cls parameter to click.option decorator like:
@click.option('--username', prompt=True, cls=NotRequiredIf,
              not_required_if='authentication_token')
How does this work?
This works because click is a well designed OO framework. The @click.option() decorator usually instantiates a click.Option object but allows this behavior to be overridden with the cls parameter. So it is a relatively easy matter to inherit from click.Option in our own class and over ride the desired methods.
In this case we over ride click.Option.handle_parse_result() and disable the need to user/password if authentication-token token is present, and complain if both user/password are authentication-token are present.
Note: This answer was inspired by this answer
Test Code:
@click.command()
@click.option('--authentication-token')
@click.option('--username', prompt=True, cls=NotRequiredIf,
              not_required_if='authentication_token')
@click.option('--password', prompt=True, hide_input=True, cls=NotRequiredIf,
              not_required_if='authentication_token')
def login(authentication_token, username, password):
    click.echo('t:%s  u:%s  p:%s' % (
        authentication_token, username, password))
if __name__ == '__main__':
    login('--username name --password pword'.split())
    login('--help'.split())
    login(''.split())
    login('--username name'.split())
    login('--authentication-token token'.split())
Results:
from login('--username name --password pword'.split()):
t:None  u:name  p:pword
from login('--help'.split()):
Usage: test.py [OPTIONS]
Options:
  --authentication-token TEXT
  --username TEXT              NOTE: This argument is mutually exclusive with
                               authentication_token
  --password TEXT              NOTE: This argument is mutually exclusive with
                               authentication_token
  --help                       Show this message and exit.