Cross-posting from the prediction thread. Haven't read over thoughts yet.
Squire
In general comparison to Steward, Squire is definitely a bit weaker. This is reflected in the $2 price. You might buy it for the extra actions or buys if the Kingdom otherwise lacks those resources.
At first glance it looks to be superior to its fellow $2 choice card, Pawn. But one of the best uses of Pawn is non-terminal +Buy, which you cannot do with Squire.
The most interesting thing is the on-trash effect. In Kingdoms without trashers, on-trash effects are inaccessible (barring a crazy new DA rule -- trash a card every turn unless you reveal a Shelter??). When trashing is available, Squire is still pretty narrow. It's not as reliable as a Feast because you have to line it up with the trasher.
Of non-DA cards, the main ones to Squire would be Familiar and Goons. Depending on the trashers available, getting a potion is probably an easier route for Familiar. But it works so well with Goons. Goons wants heavy trashing too, so you can get down to a nice multi-Goons engine. Squire can be a moneyless way to get Goons, and it will also act as a source of Actions, and even +Buy to close out your final megaturn!
You might also pick it up in games with heavy trashing where your early economy will be terrible. You could chapel down to a thin deck and miraculously pop a few Squires into Cultists...
I think Squire will probably have quite narrow use. But in certain Kingdoms, it will be essential.
Hermit
The trashing is slow, and the benefit is a static mini-Workshop. However, trashing from the discard is, I think, better than trashing from the hand. For one thing, you don't lose a card in-hand. In the mid-to-late game, your discard pile will on average be larger than your hand as well. Therefore, I expect Hermit will be a fantastic DA-combo-enabler, picking out cards and trashing them for on-trash benefit.
One interesting (3+ card) combo is TR/KC-Hermit-Squire-(Attack card). TR/KC the Hermit, gain a Squire. Use the next Hermit to trash the Squire and gain an Attack. You can even gain another Squire to trash with the next Hermit, if you are so inclined.
Madman
Hermit is also like a delayed one-shot Tactician. You skip your buy to get a future-Tactician in Madman. Madman is a little weaker than Tactician, but still pretty good. You lose out on a +Buy, but it can have a stronger effect if you have a bit of card-drawing ability. From the wording, it looks like Madman isn't throneable. Too bad.