So in a case like that of Joey Bosa, what happens if a player gets drafted into the NFL, but never signs with the team that drafts him? How long does the team retain the rights to him?
Keeping with your example of Bosa.
If he continues to refuse to sign with the Chargers, he cannot sign or play with any other NFL team this season. (I also think they have closed the loophole that allowed them to go play in Canada.)
The Chargers, by drafting him, have exclusivity to his 'signing rights' for the season. They can try to trade those rights to another team, but that is highly unlikely since NFL rookies are capped at what they can be paid, they are squabbling over "guaranteed" money at this point. No other team wants this headache going into their season.
IF, Bosa remains unsigned OR the rights to sign him traded by the Chargers throughout the year, he will be re-entered into the NFL draft.
Bosa won't do this because the average NFL career is just under 3 years (2.5 actually) and by sitting out and not signing, he is not getting paid, and eating up one of those years (sitting out etc. doesn't extend the avg career). YES, he is not playing, and not getting hit etc. but as they say 'There is no practice for playing in the NFL other than playing.'