Are you asking for a proof of the "fact" that all proofs of the theorem about cubic surfaces having 27 lines are non-trivial? If so, then no, there is not even a definition of non-trivial proof, let alone a proof that any specific proof is non-trivial or a proof that there exists a fact all of whose proofs are non-trivial.

My definition of a trivial proof would be a proof that is mechanically verifiable. I.e., it is verifiable by even something as "dumb" as a computer.

Of course, such proofs tend to be longer than most proofs that are considered non-trivial.

Maybe years ago that would have been a plausible definition, but by now many proofs have been verified by computer, including several which probably no-one would feel comfortable calling trivial.

Yes, that is what I was asking for.

What does the sentence mean then?

One particular proof (of which we do not know if it is exclusive) is non-trivial (whatever that means)? Or, perhaps, all known (to the author) proofs are non-trivial?

Baez didn't mean that as a mathematical statement rather as a statement in the field of history and human psychology: "So far nobody has been able to devise a proof that is widely regarded by people who study it as being trivial (where triviality of a proof is a subjective judgment for which there is (1) no formal criterion, but also (2) a surprising amount of agreement)."

>No matter how one goes about it, the proof of this theorem is nontrivial.

...I'm going to guess he really means:

"No matter how one goes about it, this proof of the theorem is nontrivial."

...and not...

"No matter how one goes about it, all proofs of this theorem are nontrivial."

More like:

"I am aware of several proofs of this theorem. None of them is trivial."