Draw ASCII boxes in boxes

Jelly,  20  19 bytes

-1 byte using the quick ` to avoid a link, as suggested by Erik the Outgolfer.

H»þ`€Ḣ>ЀHSUṚm€0m0Y

A full program taking a list [a,b,c] printing the boxes using a:2 b:1 c:0
... in fact, as is, it will work for up to 10 boxes, where the innermost box is 0 (for example).

Try it online!

How?

H»þ`€Ḣ>ЀHSUṚm€0m0Y - Main link: list of boxes, B = [a, b, c]
H                   - halve B = [a/2, b/2, c/2]
    €               - for €ach:
   `                -   repeat left argument as the right argument of the dyadic operation:
  þ                 -     outer product with the dyadic operation:
 »                  -       maximum
                    - ... Note: implicit range building causes this to yield
                    -       [[max(1,1),max(1,2),...,max(1,n)],
                    -        [max(2,1),max(2,2),...,max(2,n)],
                    -        ...
                    -        [max(n,1),max(n,2),...,max(n,n)]]
                    -       for n in [a/2,b/2,c/2]
     Ḣ              - head (we only really want n=a/2 - an enumeration of a quadrant)
         H          - halve B = [a/2, b/2, c/2]
       Ѐ           - map across right with dyadic operation:
      >             -   is greater than?
                    - ...this yields three copies of the lower-right quadrant
                    -    with 0 if the location is within each box and 1 if not
          S         - sum ...yielding one with 0 for the innermost box, 1 for the next, ...
           U        - upend (reverse each) ...making it the lower-left
            Ṛ       - reverse ...making it the upper-right
             m€0    - reflect €ach row (mod-index, m, with right argument 0 reflects)
                m0  - reflect the rows ...now we have the whole thing with integers
                  Y - join with newlines ...making a mixed list of integers and characters
                    - implicit print - the representation of a mixed list is "smashed"

Python 2, 107 103 bytes

a,b,c=input()
r=range(1-a,a,2)
for y in r:
 s=''
 for x in r:m=max(x,y,-x,-y);s+=`(m>c)+(m>b)`
 print s

Full program, prints boxes with a=2,b=1,c=0

Slightly worse answer, with list comprehension (104 bytes):

a,b,c=input()
r=range(1-a,a,2)
for y in r:print''.join(`(m>c)+(m>b)`for x in r for m in[max(x,y,-x,-y)])

Charcoal, 14 bytes

Ｆ#*@ＵＯ÷Ｎ²ι‖Ｃ←↑

Try it online! Link is to verbose version of code.