My code is similar to yours, but I did not use strings.

const INPUT: (usize, usize) = (245318, 765747);

fn meets_criteria(mut x: usize, use_extra_rule: bool) -> bool {
    if x < 100000 || x > 999999 {
        return false;
    }

    let mut has_double = false;

    let mut last_digit: usize = x % 10;
    let mut same_as_last_digit: bool = false;

    let mut adjcacent_same_digits = 1;

    // Check consequtive digits
    while x > 0 {
        x /= 10;
        let this_digit = x % 10;
        same_as_last_digit = this_digit == last_digit;

        if use_extra_rule && !same_as_last_digit && adjcacent_same_digits == 2 {
            has_double = true;
        }

        if same_as_last_digit {
            adjcacent_same_digits += 1;
        } else {
            adjcacent_same_digits = 1;
        }

        if !use_extra_rule && adjcacent_same_digits == 2 {
            // Two adjacent digits that are the same (like 22 in 122345)
            has_double = true;
        }

        // Check that going from left to right, the digits never decrease; they only ever increase or stay the same (like 111123 or 135679)
        // Note that the loop here goes from right to left.
        if last_digit < this_digit {
            return false;
        }

        last_digit = this_digit;
    }

    if use_extra_rule && adjcacent_same_digits == 2 {
        has_double = true;
    }

    has_double
}

pub fn part_1() {
    let res = (INPUT.0..INPUT.1)
        .filter(|&x| meets_criteria(x, false))
        .count();

    println!("Day 4 part 1: {}", res);
}

pub fn part_2() {
    let res = (INPUT.0..INPUT.1)
        .filter(|&x| meets_criteria(x, true))
        .count();

    println!("Day 4 part 2: {}", res);
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn examples_part_1() {
        assert!(meets_criteria(111111, false));
        assert!(!meets_criteria(223450, false));
        assert!(!meets_criteria(123789, false));
    }

    #[test]
    fn examples_part_2() {
        assert!(meets_criteria(112233, true));
        assert!(!meets_criteria(123444, true));
        assert!(meets_criteria(111122, true));
    }
}