Am I eligible for the American Opportunity Credit if I paid all of my childs tuition from Coverdell funds?

1066R? Do you mean 1099-Q, that's the form you get from a Coverdell. 

Yes, you do get to claim the American Opportunity Credit, but then some of the box 2 (of the 1099-Q) amount will be taxable.

Total qualified expenses (including room & board) less amounts paid by scholarship less amounts used to claim the Tuition credit equals the amount you can use to claim the earnings exclusion on the 1099-Q. 
Example:
  $10,000 in educational expenses(including room & board)

   -$3000 paid by tax free scholarship

   -$4000 used to claim the American Opportunity credit

 =$3000 Can be used against the 1099-Q 

  Box 1 of the 1099-Q is $5000

Box 2 is $600

3000/5000=60% of the earnings are tax free

60%x600= $360

You have $240 of taxable income (600-360)