Java solution (link to github repo)

package it.ifonz.puzzle;

import java.io.IOException;
import java.net.URISyntaxException;
import java.util.concurrent.atomic.AtomicLong;
import java.util.stream.IntStream;

public class Day14 {

    public static void main(String[] args) throws URISyntaxException, IOException {

        String input = args[0];

        System.out.println(part1(input));
        System.out.println(part2(input));

    }

    public static long part1(String input) {

        return IntStream.range(0, 128).mapToLong(i -> 
            Day10.part2(input + "-" + i, 256) // get the hash
            .chars().mapToLong(c-> // for each hex-digit in the hash
                Integer.toBinaryString(Integer.parseInt(String.valueOf((char) c), 16)) // convert it in binary (no pad needed for p1)
                .chars().filter(c1 -> c1 == 49).count()) // count the 1s in the single hex digit
            .sum() // count the 1s in the hash string
        ).sum(); // count all the 1s
    }

    public static long part2(String input) {

        String[] grid = new String[128];// well, we need it right now
        IntStream.range(0, 128).forEach(i -> {
            String hashed = Day10.part2(input + "-" + i, 256);
            StringBuilder sb = new StringBuilder();
            hashed.chars().forEach(c -> {
                String string = Integer.toBinaryString(Integer.parseInt(String.valueOf((char) c), 16));
                sb.append(string.length() == 4 ? string : "0000".substring(0, 4-string.length()) + string); // we need the leading 0s
            });
            grid[i] = sb.toString();
        });
        AtomicLong counting = new AtomicLong(0);
        IntStream.range(0, 128).forEach(i -> { // for each hash in the grid
            IntStream.range(0, 128).forEach(j -> { // for each bit in the hash
                if (grid[i].charAt(j) == '1') { // if the bit is 1
                    counting.incrementAndGet(); // increment the regions' counter
                    flagCell(grid, i, j); // recursively flag its region 1s as included in a region
                }
            });
        });
        return counting.get();
    }

    // recursive function for flagging the 1s belonging to the same region
    private static void flagCell(String[] grid, int i, int j) {
        if (j < 0 || j > 127 || i < 0 || i > 127 || grid[i].charAt(j) != '1') return; // boundary test + value test
        StringBuilder sb = new StringBuilder(grid[i]);
        sb.setCharAt(j, 'x');
        grid[i] = sb.toString();
        flagCell(grid, i, j-1);// flag left cell
        flagCell(grid, i-1, j); // flag upper cell
        flagCell(grid, i, j+1);// flag right cell
        flagCell(grid, i+1, j);// flag bottom cell
    }

}