Raging while in Wild Shape
Wild Shape explicitly allows this. From PHB p 67:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
If a bear is physically incapable of flying into a primal Rage, then we are doing something very wrong as a society.
Using Wild Shape while in a rage
The only thing you cannot do while in a Rage is cast or concentrate on spells (PHB p 48). Although Wild Shape is a "magical transformation", it is never described as a spell and does not require concentration.
However, a Rage ends early if (PHB p 48)
...you haven't attacked a hostile creature since your last turn or taken damage since then.
This is a problem if you aren't Circle of the Moon (and what Bearbarian wouldn't be?), as vanilla Wild Shape takes an action, leaving you dependent on opportunity attacks or taking damage to sustain your rage.
Rage Damage with Natural Weapons
One of the features of Rage is a bonus to your damage while raging. This feature states that you can only apply if when making a melee weapon attack using Strength:
When you make a melee weapon attack using Strength, you gain a bonus to the damage roll ...
All melee weapons available to PCs are Strength-based, some have the finesse property. From PHB p 147:
...you use your choice of your Strength or Dexterity modifier for the attack and damage rolls. You must use the same modifier for both rolls.
Similarly, the damage bonus on natural weapon attacks is almost always equal to either the Strength or Dexterity ability modifier (there are exceptions for CR 0 creatures). Thus, it appears every melee natural weapon attack can be Strength based, although not necessarily using the published attack/damage bonuses.
Consider the Blood Hawk (MM p 319). Strength 6 (-2), Dexterity 14 (+2), its beak has an attack bonus of +4 and a damage bonus of +2. This is certainly consistent with its beak being a finesse weapon that is using its Dexterity score. A Raging Blood Hawk (what a great mascot name!) could use its beak as a Strength-based attack, with an attack bonus of +0 and a damage bonus of -2, and then add Rage Damage.
More generally, if your Strength modifier matches the published damage bonus, the attack is Strength-based. If the Dexterity modifier matches, it's a finesse attack, and you would need to recalculate the attack and damage modifiers bonuses using your Strength bonus to turn it into a Strength-based attack.
Best Answer
Yes this will work but you have to check the monster's statblock
TLDR
These features work together; this is because Reckless Attack does not require the Attack action but only an attack and attacks made will benefit from Reckless Attack because they are "melee weapon attack rolls" that are in the case of most beasts "using Strength".
Reckless Attack states:
Notably this does not require you to take the Attack action, but only make an attack (See this Q/A if the distinction is confusing ("What does upper-case-A-Attack action vs. lower-case-a-attack mean?")). It also only benefits "melee weapon attack rolls using Strength".
The Multiattack feature on the giant badger monster states:
Note, there is no order specified for the attacks (it does not say "one with its bite and then one with its claws) so you could use them in any order.
This is supported by Jeremy Crawford (lead game designer) in his tweet here so it is certainly the rules as intended:
Clearly this was under the assumption that the DM is controlling the creature, and so the PC using Multiattack would likewise be able to choose the order of its attacks.
The badger's bite and claw attacks are both classified as "Melee Weapon Attacks". And the Player's Handbook's section on "Modifiers to the Roll" it states:
Unfortunately, this does not mean any monster with a melee weapon attack definitely uses Strength.
An example and explanation is below:
For example, the eagle's Talons attack has a +4 to hit even though its strength has a -2 modifier. The Eagle seems to be using Dexterity for its melee weapon attack. However, there is no explicit mention of this in any statblock, so the decision is ultimately up to the DM, though they can make an educated guess as to what the intention of the designer was.
You would have to determine what ability score modifier a creature is using for its attack, this can be done by first determining its proficiency bonus (look at a skill it is proficient in and subtract the ability score modifier normally associated with that skill to get the monster's proficiency bonus).
Then look at its bonus for the to hit part of that attack to see how much is added (or subtracted) from the proficiency bonus.
If a monster has equal strength and dexterity modifiers I would ask your GM to decide what is used.
Of note: some monsters are not proficient in all of their attacks and so they may not be adding their proficiency bonus to the to hit modifier.
An example of this is laid out in the Sage Advice Compendium:
When looking at the statblock runs into issues just ask your GM, you can even ask them before you get into the statblock and just use what makes sense to the GM.