Is what the religious young man does—who relies on his family due to his illness—considered blameworthy reliance (tawakkul), and will someone who relies blameworthily enter Hell eternally?

If a person is ill and unable to earn, he is not described as relying on others (mutawakil), and his maintenance is obligatory upon his able father. As for someone who abandons earning while being capable of it, he is indeed relying on others. However, not everyone who abandons an obligation incurs the threatened punishment; for Allah is Forgiving and Generous, and repentance erases what came before it, and eternal damnation in Hellfire is specific to those who die upon disbelief.