Yes, it will, with a caveat.
Fruit Juice interacts with your maximum HP. Whenever you increase your maximum HP in Slay the Spire, you also gain that much current HP.
For ease of understanding (simpler numbers) I assume you have the 'Potions give double' effect from Sacred Bark and 60/100 HP. That means this sequence of actions:
0. Start (HP 60/100)
1. Rest (HP 90/100)
2. Drink (HP 100/110)
Can give a different result from:
0. Start (HP 60/100)
1. Drink (HP 70/110)
2. Rest (HP 103/110)
Those few extra HP points mean it is generally a good idea to save your Fruit Juice until you run out of slots or encounter a campfire. Other reasons include having relics that interact with your health percentage. Gain strength (Red Skull) or HP (Meat on the Bone) at end of combat if health is below 50% are two examples. When you drink Fruit Juice, your HP percentage goes up. (In the example, from 60% to 63.6%). This may mean you miss out on the effect.
The caveat comes with the game's programming: if you're too fast clicking between the map, camp screen, and card reward screen (there is a relic that gives you cards at the campfire) then the interaction can jumble up with certain sequences and produce inconsistent and even otherwise impossible results, such as using a potion twice or not losing it when using it, or not getting any effect at all. Most regular players consider this kind of glitchy technique cheating.