I would agree that there's tons of evidence that Jon isn't Ned's from the show.
'Not Ned's way', 'left and came back with a baby boy, never said from where', 'the next time we meet I'll tell you about your mother' etc. Those hints have been blatant.
But why is it supposed to be Lyanna's? We know fuck all about her other than what we got in this episode, really. I guess that would explain why Ned went and took a child that wasn't his (even though it upset his wife massively) but what else matches up? When was Jon born? Before or after Robert's rebellion? How old is he?
If he's the same age as the rebellion is old then I can understand maybe Lyanna siding with Rhaegar and the Targs, getting killed, and Ned taking the kid to spare it death. But I don't feel like we've actually had any characters say any of that and thus still feel like Jon could be anyone's kid.
He's actually a