Okay, I got my hands on a copy and have read it. Surprisingly, there's not a lot of direct comparison between Dzogchen and Zen; mostly, it's an elucidation of how Dzogchen is not a path of renunciation or transformation (and there's a lot of comparison with Tantra). But these quotes stick out most as the essence regarding Dzogchen and Zen:
Thus, one can understand that the principle method of Zen is a
way to find yourself in the absolute condition. This principle is a
common element between Chinese Buddhism and Zogqen. But
you must not therefore think they are one and the same thing.
You must never forget that the two methods are different. We
have already spoken of one as the way of self-liberation and the
other as the path of renunciation. From the beginning, in
principle, these two methods are very different.
To my naive ear, it doesn't sound like Rinpoche is differentiating the end state (the absolute condition) between the two, but distinguishing their methods.
As for the ways in which Chinese Buddhism (Ch'an) is a path of renunciation, there's not much said, except for this section a few pages earlier:
In the Buddhist sutras, fundamentally the method used is that
of renunciation. This is true of all of the different schools of
sutric Buddhism, whether it be Theravada, or the school of Zen
Buddhism, or the Mahayana Buddhism of the followers of
Nagarjuna. In this approach, the first action, the first gesture of
those who present themselves as Buddhists is, for example, to
take refuge. When we take refuge, there is, above all,
something negative which we have renounced. It is sufficient,
for example, to think about the fact that in taking refuge one
receives a name. Why does one have to receive a name, if you
are already born with a name? This is a way of expressing, of
exemplifying, that one is renouncing one's prior life and how
one was. If one goes a step further, the best way of living in this
life is held to be monkhood; if one wishes to truly sustain and
support the teaching, one should become a monk, it is said. If
one gives such importance to being a monk, what is the reason?
The monk is the example of renunciation of all that is worldly.
In regard to overcoming all kinds of negativity, the method here
is one of renunciation through the application of an antidote,
such as compassion applied to hatred.
There are certainly many contemporary Dzogchen masters who emphasize going for refuge, and according to the above, presumably they would be considered as teaching paths of renunciation as well.
All in all, very interesting.