I think the first version may be interesting. I would anticipate it feeling more like crazyhouse and would be more susceptible to being solved, but may be worth trying. The second version seems unworkable. In chess, time changes the speed of the game, but doesn't affect the game. There is no benefit to being up on time other than having more time to think. In bughouse, time is an advantage on the board. Both sides effectively have a piece generator that can be turned on and off, and both sides have tempi considerations across boards. The reason people don't play long bug games is because when time is tight, a typical strategy is to sit the clock down -- if you have 20 minutes, you sit for 18 minutes; if you have 3, you sit for 1.
Correspondence bughouse

Agree with Antisomniac, the second version is unworkable. About the first one, it can be interesting (I actually proposed it as well) but of course it becomes a complete information game (and this means it's solvable, with a very very complex move tree), while bughouse of course it's not solvable because it's an uncomplete information game... and that's the beauty of the game. Also, the first version would be easy to play 1 vs 1, while simuling in bughouse costs time
There is simply no way to make correspondence bughouse work in the same way, with exactly the same strategies, as live bughouse. However, here are a couple ideas that could loosely mimic the feel of live bughouse in a correspondence setting.
First idea: Enforced Move Order
You have an enforced move order: W1, W2, B1, B2. White moves on the first board, then white moves on the second board. Then black moves on the first board, then black moves on the second board. Alternatively, you could do W1, W2, B2, B1, but I don't think there's any significant difference between them.
However, sitting to wait for material is an important part of the game. An enforced move order eliminates this. So what we could do is add the ability to pass on your turn a limited number of times; every time you sit on your turn, it takes one of your remaining sits and gives it to your opponent.
So if everyone gets two sits, if W1 sits once, he'll have one left, and B1 will have three.
This could lead to the following sequence of moves:
W1, W2 sits, B1, B2 is skipped, W1, W2, B1, B2.
Second idea: Super-long Live Time Controls
Instead of having 3 days per move, instead have three days total with a 12-hour (or 24-hour) increment. That way, sitting actually does eat time that may cost you later in the game, just like in live. Perhaps the total time should be capped so that you can never have more than 7 days left on the clock so that you can't sit for two months while waiting for your opponent to grab that knight.
If anyone goes on vacation, either it could not apply to the bughouse game, or it would have to stop the clock and disable moves on both boards.
Both options have their drawbacks. And there's a significant drawback to any form of correspondence bug that I don't think has a solution: the stronger player can simply tell his weaker partner what to do and will therefore be effectively self-simuling. This can theoretically be done in live too, but fast time controls make it difficult, and it will always result in lowering the quality of play. In correspondence, that is no longer the case.
This last issue is the one that means that correspondence bug is never really going to take off and become particularly popular, as it effectively destroys it as a team game.