I think 'you' in general statements, as in, say, 'You need to follow the traffic rules', is singular. Then about this:
Even if someone you love becomes sick and he or she may die, it's not your fault and there are times when there is nothing you can do. You don't always have everything under control.
is it possible to replace 'he or she' above with 'they' even though practically your loved one is a single person?
Yes, they is commonly used as a gender-neutral singular pronoun.
In your sentence, however, no pronoun is necessary.
'If someone you love becomes sick and may die. . . .'