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Perl, 263 bytes

Takes input as 64 character string of 1s and 0s.

```
perl -E'@g=map[split//],unpack"(A8)8",$_=<>;$i=index$_,"0";$I=index$_,"1";sub c{my($x,$y)=@_;$x<0||$x>7||$y<0||$y>7||$s[$x][$y]++||$g[$x][$y]-$c?0:1+c($x+1,$y)+c($x,$y+1)+c($x-1,$y)+c($x,$y-1)}$c=0;$r=c(int($i/8),$i%8);@s=();$c=1;$r+=c(int($I/8),$I%8);exit 64-$r'
```

Comments:

```
perl -E'
    # load input into array
    @g = map[split//], unpack"(A8)8", $_=<>;

    # locate first 0 and 1
    $i = index$_,"0";
    $I = index$_,"1";

    # recursive connectivity search
    # returns count of connected elements
    sub c{
        my ($x,$y) = @_;

        $x<0||$x>7||$y<0||$y>7  # bounds check
        || $s[$x][$y]++ ||      # already visited?
        || $g[$x][$y]-$c        # correct value?
        ? 0
        : 1 + c($x+1,$y)+c($x,$y+1)+c($x-1,$y)+c($x,$y-1)  # recurse
    }

    # count connected 0s/1s
           $c=0; $r  = c(int($i/8),$i%8);
    @s=(); $c=1; $r += c(int($I/8),$I%8);

    # sum of connected 0s and 1s should total 64 for success
    # non-zero exit code means failure
    exit 64-$r
'
```