What does the phrase, “Eat your heart out, Samuel Becket” in reference to Republican Presidential candidate debate?

Solution 1:

The word 'soothsayer', though commonly used to mean 'future teller', originally meant 'truth-teller'. This is the sense in which the article means it - that there are many people 'telling the truth' (or at least saying that they are telling the truth, because they contradict each other).

Becket's work was often about the nature of truth, and often featured characters who claimed to tell the truth but maybe didn't. The implication is that even in his most Absurdist moments he never had as many contradictory characters claiming to tell the truth as the Republican Primary debates.

Solution 2:

The "eat your heart out, Samuel Becket[t]" is not directly related to the sense of "soothsaying" -- DJClayworth's answer gives a pretty good explanation of the latter term, but I think overemphasises the "truthiness" aspect of the whole thing.

It's important to note the reference to the transcript of the debate --- the line quoted from the transcript (including its attribution to the 'character' of "UNIDENTIFIED MALE") looks like a line from the script of a Beckett play. It comes out of nowhere (or at least from an 'unidentified' source), yet seemingly has some great moral weight or importance; it is the fact that such non sequitur dialogue is thought to be typical of Beckett's works that the author is making reference to. You are right that the overall idea that the author was trying to convey was that the scenario was as absurd as a Beckett play, but this is introduced through a much more direct observation of the fact that the actual transcript itself looked like a script of such a play.

As noted, Samuel Beckett was Irish.