If you are talking about the unarmed strike mentioned on page 195 of the PHB the answer is no unless the wild shaped form is a humanoid like a Ape (CR 1/2). Even then the Ape has a superior attack with his Fist.
The exact wording of unarmed strike is
When you are unarmed, you can fight in melee by making unarmed strike, as shown in the weapon table in chapter 5.
Looking at the weapon table we see
Unarmed Strike Cost:nil, Dmg: 1 bludgeoning, Weight:nil, Properties:None
It is clearly representing a humanoid ability to punch or kick. Not claws, hooves, or any of the other attacks that animals are capable of performing.
I sense there is something more to this question. But since I don't know what that is I will end my answer with the following.
From a in-game standpoint all natural animal strikes are unarmed in that they are attacks that are intrinsic to the creature and not as a result of a tool, like a sword, or a supernatural power. In this sense the Bite attack or the Claw attack of a panther is an unarmed strike using a different set of stats then the ones found on the weapons chart.
Looking at the Ape again we see that the Fist is clearly a unarmed strike of the this type. But the second type of attack is Rock which is not an unarmed strike of any kind as it involves the ape picking up a rock and throwing it. So a druid, wild shaped as an ape, is attacking unarmed when using the multi-attack with two fists.
To make a better answer I need to know why it is important to make the distinction that a wild shaped Druid is making an unarmed strike.
The Druid/Monk Multi-Class character
In my opinion the literal reading of the rules makes my answer no. The unarmed strike that various PHB abilities refers to is specific weapon listed on the weapon table in Chapter 5 of the PHB. While various animals are certainly attacking unarmed they are not attacking with that weapon which is limited to the PHB races of humanoid form.
However if you want to go with the common sense idea that a animal's natural attack is an unarmed strike. Then I would consider the following:
The main issue is the increased damage for unarmed strike. A normal human does 1 point of damage. The monk starts off with 1d4 and eventually increases to 1d10. I would say this does not apply while in a wild shape. The idea of increased damage is the monk overcoming the limitation of human punches and kicks through skill and training. A limitation that animals with claws, fangs, and hooves do not possess.
If the referee feel strongly that a monk training would increase the damage of an animal's natural attack then he needs to figure out a way to stepping up the attack die in a progression similar to the monk table. I feel that the this was not the designers intent.
As for the other abilities of a Monk, I would look careful at the description of the ability and see how the use of humanoid hands are involved. For example I don't see any particular issues with Flurry of Blows, however Deflect Missiles a Druid in a wild shape without hands would not be able to make use of the ability to catch an arrow, deflection would be unaffected. If the feels that it important to retain this ability he could add a house rule that it transform into the ability to catch the arrow with form's mouth or other suitable appendage like a tentacle if applicable.