Every one keeps making jokes about how dumb it was from Jon to shake Jude’s hand and how he should have expected to get burn but what else could he do?

You can hear how hesitant he is to agree he knows it’s a trick but also he doesn’t see any other option to get what he want.

Because, you know, Jude insisted that she will give Jon location of Mike Crew only if he shakes her hand. If he refused and called her out she would probably shrug and walk away not telling him what he wanted.

You think if Sasha was the Archivist she wouldn’t shake her hand to get the information? She literally went to the location Michael told her to because it was only way to get information about a thread they were all facing she would definitely shake Jude’s hand as well.