This answer has since been invalidated. It was originally based upon making a call using insufficient information. Jeremy Crawford has now posted clarification that the Hand has no Speed characteristic and simply moves magically. I am leaving it up to allow other DMs to have support of argument should they wish to rule differently in their own games. Also, The part about 3 dimensional movement stands
Yes, but it's not quite as good as you describe.
Under Moving a Grappled Creature:
When you move, you can drag or carry the grappled creature with you,
but your speed is halved. unless the creature is two or more sizes
smaller than you.
So the hand's speed (as a Large "creature") is likely to be 30 feet, not 60, unless the target is Small or smaller.
A case could be made for the full movement, as the hand does not have a Speed, but is an effect that can be moved. Since the hand has an AC, Strength, Dexterity and Hit Points, it's probably fair to rule that the movement is governed by by the same rules as Speed. The rules are ambiguous as to whether the movement is allowed in 3 dimensions, but since it is a magical hand of force that is not described as walking around on its digits, flight or hovering must be assumed. I would lean towards full three dimensional movement, but in the absence of clear description, a DM could rule otherwise.
When you cast the spell and as a bonus action on your subsequent
turns, you can move the hand up to 60 feet and then cause one of
the following effects with it.
This is a specific case which overrules general. The move must come before the effect.
So you could move the hand to the target and make the grapple.
Then, on you next turn, you could move and crush. The rules state this takes a bonus action to have it perform the crushing, but it does not specify if this is the same bonus action you used to move and make an allowed action, or an alternate. Again, the wording makes this frustratingly unclear. If it is the same, you could do the crushing damage, if not, you would have to forgo the crushing damage to move the hand.
The target gets another breakout roll on its turn. Failing that, you could move the hand another 30 feet and/or crush again (depending on your DM's call)
In short, yes, it should be possible, but the Hand will move only half speed (starting the subsequent turn after the Grapple) unless you are grabbing a Small or smaller creature, and your DM might disallow the crushing damage. Also, the grabbed person will be able to attempt a breakout contest on each of their turns.