##AWK, 90 bytes

{z=$1*$1
for(x=$1;x>=0;x--)for(y=0;y<=$1;y++){d=z-x*x-y*y
if(d>0&&d<2*(x+y)+2)c++}$0=4*c}1

Usage:

awk '{z=$1*$1
    for(x=$1;x>=0;x--)for(y=0;y<=$1;y++){d=z-x*x-y*y
    if(d>0&&d<2*(x+y)+2)c++}$0=4*c}1' <<< 5525

Just a simple search through quadrant 1 to find all boxes that will intersect the circle. Symmetry allows for the multiply by 4. Could go from -$1 to $1, but that would take a more bytes and be less efficient. Obviously this is not the most time efficient of algorithms, but it only takes about 16 seconds to run the 5525 case on my machine.

AWK, 90 bytes

{z=$1*$1
for(x=$1;x>=0;x--)for(y=0;y<=$1;y++){d=z-x*x-y*y
if(d>0&&d<2*(x+y)+2)c++}$0=4*c}1

Usage:

awk '{z=$1*$1
    for(x=$1;x>=0;x--)for(y=0;y<=$1;y++){d=z-x*x-y*y
    if(d>0&&d<2*(x+y)+2)c++}$0=4*c}1' <<< 5525

Just a simple search through quadrant 1 to find all boxes that will intersect the circle. Symmetry allows for the multiply by 4. Could go from -$1 to $1, but that would take a more bytes and be less efficient. Obviously this is not the most time efficient of algorithms, but it only takes about 16 seconds to run the 5525 case on my machine.