All Stark Children show the gift of Skin Changing as GRRM confirmed.
Jon and Bran are described as particularly powerful Skinchangers. Bran is even more special because he is the Winged-wolf while his siblings aren't.
Jon has shown his abilities (Unintentionally) as a warg on following occasions:
- In controlling his wolf. There is no such thing as a tame direwolf.
- Spying on Wildling Army in Frost Fangs while on a ranging with Qhorin Halfhand. Jon was asleep and Ghost was miles away, yet Jon managed to slip into the skin of his wolf and see what was waiting ahead of them.
- He slipped into Ghost's skin after the Battle of Castle Black when he was standing beyond the wall and Ghost came back from the wild.
Jon warged into his wolf to scout the Wildling army albeit unintentionally.
And suddenly he was back in the mountains, his paws sunk deep in a
drift of snow as he stood upon the edge of a great precipice. Before
him the Skirling Pass opened up into airy emptiness, and a long
vee-shaped valley lay spread beneath him like a quilt, awash in all
the colors of an autumn afternoon.
A vast blue-white wall plugged one end of the vale, squeezing between
the mountains as if it had shouldered them aside, and for a moment he
thought he had dreamed himself back to Castle Black. Then he realized
he was looking at a river of ice several thousand feet high. Under
that glittering cold cliff was a great lake, its deep cobalt waters
reflecting the snowcapped peaks that ringed it. There were men down
in the valley, he saw now; many men, thousands, a huge host.
A Clash Of Kings: Chapter 53
Jon's comrades when scouting the Skirling Pass were the first to realize Jon was a warg after he told them about his "dream".
It made him feel half a fool to talk of such things to Qhorin and the
other rangers, but he did as he was commanded. None of the black
brothers laughed at him, however. By the time he was done, even Squire
Dalbridge was no longer smiling.
“Skinchanger?” said Ebben grimly, looking at the Halfhand. Does he mean the eagle? Jon wondered. Or me? Skinchangers and wargs belonged
in Old Nan’s stories, not in the world he had lived in all his life.
Yet here, in this strange bleak wilderness of rock and ice, it was not
hard to believe.
“The cold winds are rising. Mormont feared as much. Benjen Stark felt
it as well. Dead men walk and the trees have eyes again. Why should
we balk at wargs and giants?”
A Clash Of Kings: Chapter 53
Later Jon accepted it as well and started to understand it.
it was a hunger inside him, sharp as a dragonglass blade. A hunger...
he could feel it. It was food he needed, prey, a red deer that stank
of fear or a great elk proud and defiant. He needed to kill and fill
his belly with fresh meat and hot dark blood. His mouth began to water
with the thought.
It was a long moment before he understood what was happening. When he did, he bolted to his feet. “Ghost?” He turned toward the wood,
and there he came, padding silently out of the green dusk, the breath
coming warm and white from his open jaws. “Ghost!” he shouted, and the
direwolf broke into a run.
A Storm Of Swords: Chapter 79
Is Jon aware of his power?
Jon is aware that he is a Warg. He was told that by the Wildlings when he went beyond the Wall with the Old Bear and the rest of the Night's Watch on the Great Ranging. Later he was taunted as "Beast boy" and "Warg" by Night's Watch Men who wished him ill. But Jon is untrained in the art, unlike Bran who is trained by the Three Eyed Raven himself.
Not only did Jon know it, the Wildings knew it, many people in the Watch knew it and so do some Northern aristocrats such as Cregan Karstark:
Cregan: I see what you are, Snow. Half a wolf and half a wildling,
baseborn get of a traitor and a whore. You would deliver a highborn
maid to the bed of some stinking savage. Did you sample her yourself
first? If you mean to kill me, do it and be damned for a kinslayer.
Stark and Karstark are one blood.
As Jon notes in ASOS, Men of the NW call him a warg:
“What everyone knows is that Ser Alliser is a knight from a noble
line, and trueborn, while I’m the bastard who killed Qhorin Halfhand
and bedded with a spearwife. The warg, I’ve heard them call me.
How can I be a warg without a wolf, I ask you?” His mouth twisted. “I
don’t even dream of Ghost anymore. All my dreams are of the crypts, of
the stone kings on their thrones.
Also, Janos Slynt declared him for a Warg in front of everyone in the Night's Watch elections for the new Lord Commander.
Lord Janos started sputtering again. “This, this is an outrage. We
ought to hang this boy. Yes! Hang him, I say, hang him for a
turncloak and a warg, along with his friend Mance Rayder. Lord
Commander? I will not have it, I will not suffer it!”
A Storm Of Swords
Slynt also said:
Lord Janos was red-faced and quivering. “The beast,” he gasped. “Look!
The beast that tore the life from Halfhand. A warg walks among us,
brothers. A WARG! This... this creature is not fit to lead us! This
beastling is not fit to live!”
Wildling Skinchanger, Varamyr, knew it as well.
He had known what Snow was the moment he saw that great white direwolf
stalking silent at his side. One skinchanger can always sense
another. Mance should have let me take the direwolf. There would be
a second life worthy of a king. He could have done it, he did not
doubt. The gift was strong in Snow, but the youth was untaught, still
fighting his nature when he should have gloried in it.
A Dance With Dragons: Prologue
Wildlings who saw Jon kill Qhorin also said this in front of him:
“The lad’s a warg, or close enough,” put in Ragwyle, the big
spearwife. “His wolf took a piece o’ Halfhand’s leg.”
The Weeper’s red rheumy eyes gave Jon another look. “Aye? Well, he has
a wolfish cast to him, now as I look close. Bring him to Mance, might
be he’ll keep him.” He wheeled his horse around and galloped off, his
riders hard behind him.
Later Rattleshirt said this in front of Jon and the council of Wildling chiefs:
“The black bastard what gutted Orell,” said Rattleshirt, “and a
bloody warg as well.”
Can Bran take Jon with him?
Difference between Bran and Jon is that Jon is a skinchanger but Bran is a Greenseer. The Three Eyed Raven chose him as his successor and the Children of The Forest hinted that they were waiting for Bran for a long time. Which means Bran is either some sort of a prophesied Messiah or he is exceptionally talented. While we have seen that two Greenseers can tag along in visions but it has never been disclosed if a Greenseer can take a normal human or a common warg with him in visions.