There is an idea floating around that Illyrio's wife was a Blackfyre, who are only extinct in the male line, and that baby Aegon is the offspring of their union, or to use terms I would actually use, is their kid. Whether his wife is a Blackfyre or no, it still is convincing, Illyrio is a proper soppy bastard around this kid. I'd say JonCon is unreliable in that he is simply being duped, not lying to us the readers, because that would be impossible, obviously there is a bit of Sansa Stark history clouding in there as well, the Hound never kissed her at the Blackwater, but she certainly thinks he did. Things like that crop up from time to time. Whatever the reasons, we are on the same lines here, he believes something that isn't true.
If you like foreshadowing, check out the story of the Dragon sign on the inn, it left these shores a black Dragon but washed back up red. Bit of a reach though, I'll admit.
Also, the Golden Company is awfully convenient, add in that they were not at all surprised at JonCon's revelation and I am already thinking they know more than he does. Would they ally with Targs just because they have Blackfyre blood in them? I dunno, obviously if you've read Dunc and Egg you'll be more doubtful, since that whole shitstorm is elaborated on a bit more.
Anyway, the clues are there to support whatever you think, so if Aegon does go on to be any way successful then I doubt we'll ever know for sure, if he does for Dany or especially if she does for him, we'll never know for sure. Though I am convinced neither will take the Throne in the end, GRRM has stated that it'll be someone surprising, but I can't see Aegon getting passed Dany and Dany wouldn't be surprising.
Part of me wants him to win, and then have it revealed in a final POV (perhaps Varys with Illyrio turning on him) that the kid is just his own, and that he's planted his son on the Iron Throne and fucked up Targs, Blackfyres and Baratheons in the process.