If you shuffle your discard because it was empty to draw, does the chosen unit go back in the shuffled deck?

When you run out of cards in your Main Deck while attempting to draw, you trigger a Burn Out. This is a specific game mechanic that replenishes your deck using your trash.

According to Rule 418.2, when you Burn Out, you perform the following sequence:

1. Perform as much of the action as possible: You draw as many cards as you can (which is zero if your deck is empty).
2. Recycle your trash into your Main Deck: All cards currently in your trash are moved to your Main Deck.
3. Randomize: Per the reminder in Rule 418.2.b, when multiple cards are recycled to the Main Deck at the same time, they must be randomized (shuffled).
4. Opponent gains 1 point: You choose an opponent to gain 1 point.
5. Complete the remainder of the action: You then draw the remaining number of cards you were originally instructed to draw from your newly replenished and shuffled deck.

During a Burn Out, any unit card currently in your trash will be moved into your Main Deck and shuffled as part of the randomization process, as the process instructs you to "Recycle your trash into your Main Deck." Therefore, a unit that was in your trash does get shuffled back into your deck.