How should young leaders deal with Soldiers who have poor personal hygiene issues?

It would definitely start with a private talk with the soldier on the lowest leader level. If the soldier genuinely refuses to conduct proper hygiene then scare tactics from medical personnel would be a tool.(pictures are pretty real) if there is nothing that can motivate the soldier then counseling on failure to adapt to military lifestyle is necessary.

I had a situation that occurred with an Airman that I supervised as a SSgt/E5. Three of the other women in our office came to me with a complaint about her hygiene and more or less demanded I did something about it. I addressed their tone, and my disappointment that they couldn't take her aside as peers but ultimately, something needed to be said or done.
I did speak with her, and found that she wasn't aware of the problem or how to effectively deal with it. I asked her if she would be comfortable having someone that knew more about that sort of thing than I did, she was receptive and I made her an appointment at the Health Office.
In the end, everything worked out for the best it just took compassion and understanding.

When I was but a wee lad, my father told me that in the Army if they had a guy who didn't wash they would take him out back and scrub him with a wooden brush and the garden hose.

Of course that was at the tail end of World War II... ;-)