I tried this with GCC. And it works. Besides, even when not compiling and running this little thing you can see that it is correct. Your string[0] == 'e' and string[1] == 'e'. So there shouldn't be problems.

Code:
#include <stdio.h>

int main (void)
{
    char string[ 3 ] = "ee";

    if ( string[ 0 ] == string[ 1 ] ) 
       printf( "match" );

}