lol SG.
That is right, if it does not hurt it probably ain't broke. But it still needs recovery time... remember not to over train.
Squats are simply awesome for legs. It is pretty much all encompassing utilizing the entire leg.
It mainly targets the quads (included outer thigh) and secondarily targets the inner thigh
Inner thigh: Synergistic action - more specifically the Adductor magnus muscle - pretty much that big muscle you feel at the upper most portion of your inner thigh. (there is another muscle on top of that call the gracilis)
Outer thigh: Target - Quadraceps - This bad boy is made up of 4 muscle hence the 'Quad' portion of the name. The outer most, lateral, muscle is the vastus lateralis or externus. It is located more anterolateral. And makes up the lateral (outer) portion of your sexy thigh

If you were to place your hand smack dab (I have ever used that term before) on the side of your leg that is the tensor fascia lata..mainly tissue (fascia) and tendon (tensor)
Here are the mucsles the squat works along with their role in the movement:
Target: Quadriceps - all four
Synergists: Gluteus Maximus (butt), Adductor Magnus, Soleus (deep calf)
Dynamic Stabilizers: Hamstrings, Gastrocnemius (superfical calf)
Stabilizers: Erector Spinae (back along spine)
Antagonist Stabilizers: Rectus Abdominis (abs), Obliques
If you wanted to target the inner thigh more directly and involve the other inner thigh muscles you could try any movement inwhich your leg is brought to midline starting from it extended to the side.