I think Balon mooted that particular claim by himself:

“Quellon Greyjoy still sat the Seastone Chair when Robert Baratheon, Eddard Stark, and Jon Arryn raised their banners in rebellion. Age had only served to deepen his cautious nature, and as the fighting swept across the green lands, his lordship resolved to take no part in the war. But his sons were relentless in their hunger for gain and glory, and his own health and strength were failing. For some time his lordship had been troubled by stomach pains, which had grown so excruciating that he took a draught of milk of the poppy every night to sleep. Even so, he resisted all entreaties until a raven came to Pyke with word of Prince Rhaegar’s death upon the Trident. These tidings united his three eldest sons: the Targaryen were done, they told him, and House Greyjoy must needs join the rebellion at once or lose any hope of sharing in the spoils of victory. (emphasis mine)

Lord Quellon gave way. It was decided that the ironborn would demonstrate their allegiance by attacking the nearest Targaryen loyalists.“

(WOIAF)

The Iron Islands could not have simultaneously been independent prior to Robert’s Rebellion (Balon’s argument following his defeat at Pyke) and have joined the rebellion prior to the Battle of the Mander (Balon’s argument in 283 AC) – because that battle post-dated the news of the Battle of the Trident, and Robert’s acclamation would have been part of that news.

And this, along with so many other reasons, is why Balon Greyjoy is utter bullshit.