|The Gospel Triumphs over Heresia and the Serpent by Burchard Precht|
Short answer? No. None.
Those "c"atholics run into some series logical problems with that line of thinking. If the Church can officially teach error (which they claim it did at Vatican 2 and has continued to do since), then how can they know the Church didn't also teach error at any other time in history, including at other ecumenical councils? Maybe the Church's teaching at the Council of Trent were heretical? Then Luther might have been right about sola fide, after all. Or maybe the Council of Nicaea was teaching heresy when it declared Jesus to be consubstantial with the Father. Then Arius might have been right when he taught Jesus was merely a creature.
Further, if the Church can elect psuedo-popes (which these "c"atholics claim the Church did when it elected John XXIII), then how do we know if other popes were "true popes?" Maybe other popes were elected by nefarious secret societies too? Maybe Pius X wasn't really the pope or Pius V or Leo the Great, etc, etc, etc. Or is it only popes who challenge these "c"atholics conception of the Church that are mysterious "psuedo-popes?" An appeal to the actual anti-popes of history will do your "c"atholic friends no good here, either. For those popes were invalidly elected while another validly elected pope still reigned. These anti-popes weren't false pontiffs because they taught something wrong, they were false because someone else was already pope! Pius XII was dead. The College of Cardinals, using the rules laid down, validly elected John XXIII, therefore he was pope. Even if we grant a freemason plot had a hand in the matter (which it didn't), that still wouldn't have invalidated the election of John XXIII.
A great example from Church history proves this point nicely (a great thing about the Church having passed through 2 millennia of history is that we rarely encounter a situation that we haven't been in at least once before). We need only turn our eyes to the late 14th century to find a similar situation to the one your "c"atholic friends have dreamed up for John XXIII.
As you may recall, Gregory XI ended the exile of the papacy in Avignon, returning to Rome shortly before his death. Fearing that the next pope would be French and return the Curia to Avignon the Romans rioted upon Gregory's death. The rioting was a direct threat to the (mostly French) cardinals to elect an Italian pope. Under this duress, the cardinal's selected Bartolomeo Prignano who took the name Urban VI. Later the same year, a majority of the cardinals fled to Anagni and proceeded to elect Robert of Geneva pope! Robert, now styling himself Pope Clement VII, moved back to Avignon while Urban stayed put in Rome. Europe quickly divided itself and the Church was in for some tough times, but would survive them as she always does.
|Pope Urban VI|
As you can see, the election of Urban VI met all the conditions for a valid election (as did John XXIII's), but was "influenced" by forces outside the College of Cardinals (a Roman mob instead of a group of Freemasons). The important point for us here is that Urban VI, despite this outside influence, was the real pope. Robert of Geneva (and the Avignon "popes" following him) were anti-popes as can be seen on any official list of popes (here for example). Such "outside pressure" cannot invalidate the election of a pope. If a man is elected following the rules then in place and if the throne is vacant, then he is pope - freemason plots (and Roman rioters) notwithstanding. If this was true in the case of Urban VI (who saw his own cardinals declare his election invalid and decide to elect another man pope within a year of his coronation) how much more true would it be for John XXIII who reigned without a single cardinal claiming he wasn't the "real" pope? Ask yourself, if even the cardinals can't get a mulligan and elect a new pope (even when the first election was held under the threat of immediate physical violence), from whence do these "c"atholics derive the power to declare John XXIII an anti-pope? Which is all beside the point that there simply is no evidence at all for any "freemason plot" to elect John XXIII in the first place!
What really is at work here is an attempt to justify a rebellion against a pope (who is also a canonized saint) that these "c"atholics don't like. Nothing more.
My advice? Ignore them.