A better way to catching Raikou and Entei without a masterball
Some Pokemon with the ability Soundproof (I used Exploud since Sapphire is needed anyway)
Some way to trade between your Leaf Green/Fire Red and Ruby/Sapphire
A fast Pokemon with false swipe, preferably level 50 or lower (I used Scyther)
First, transfer the Eevee to R/S. Level it up until it learns baton pass. Then, get its friendship level up high enough to evolve it. Evolve it to Umbreon on the next level up. Continue leveling until Umbreon learns mean look. Mean look and baton pass are the only two moves Umbreon needs for this strategy, so use whatever moves you want for the other two. Once it has the needed moves, transfer it back to LG/FR.
I found in my case that I needed about 110 speed to outspeed Entei, and whatever the case is for you, Umbreon must be able to outspeed Entei/Raikou or this will not work. At level 50, Umbreon has a hard time outspeeding those two (you'll need to at least EV train for speed and possibly get a speed plus personality). Alternatively, if your Umbreon is male (it probably is), then you can breed it with a female Absol to get an Absol with those two moves. Absol has a base speed that's 10 higher than Umbreon, so it's a bit easier to get enough speed. Lastly, if you're like me and don't use/need the repel trick to find these legendaries, then just level up Umbreon beyond 50 until it is fast enough (I track them down using the Pokedex and some guesswork).
The soundproof Pokemon is much easier. You have lots of options and you can EV train and level as you see fit. You mostly only need this Pokemon to tank while you throw pokeballs. The better the defenses, though, the less often you have to heal it.
There isn't much to say about the last one. Just make sure it is fast and has false swipe.
Step 1: Weaken Raiku/Entei
Simple Way - Lead with your false swipe Pokemon. Use whatever method you like best for finding Raikou/Entei. When you get into a battle, false swipe it and let it run away. Repeat until you are sure it has only 1 HP left. Save your game. Continue to step 2.
Risky Way - Lead with Umbreon/Absol. When you get into a battle with Raikou/Entei, first use Mean Look, then Baton Pass to your false swipe Pokemon. Use false swipe constantly until Raikou/Entei is at 1 HP. This is risky, though, because Raikou/Entei can still use Roar, requiring you to reset and start over. If you're lucky enough to get it down to 1 HP using this method, then switch back to Umbreon and go to step 2. Optional: Switch to some other Pokemon and let Raikou/Entei run away. Save your game and then continue to step 2.
Step 2: Trap Raiku/Entei, and nullify Roar
Lead with Umbreon and encounter Raiku/Entei. Use mean look, then baton pass to your soundproof Pokemon.
Step 3: Catch Raiku/Entei
Throw whatever pokeball you want over and over until it works. If your soundproof Pokemon gets low on HP, heal it. If for some reason your soundproof Pokemon is knocked out, then switch back to umbreon/absol, use mean look, revive your soundproof Pokemon (hope that Raiku/Entei doesn't use roar on this turn), and then baton pass back to your soundproof Pokemon and continue as before.
No soundproof Pokemon exist that can learn mean look. It sure would have made things simpler. That's why you need an umbreon/absol + soundproof Pokemon combo.
No soundproof Pokemon exist that can learn false swipe. This means that you pretty much have to do the weakening in battle(s) before the battle you actually try to catch. The advantage of the simple way is that Raiku/Entei won't use any PP if they can run away on the first turn.
Baton pass is a bit more than a simple way to transfer stat bonuses to another Pokemon. In effect, it actually swaps the user with another pokemon without resetting anything a switch would normally reset (confusion/attraction might be an exception, but either way, that would be irrelevant to this thread). Mostly, this means that the effect of mean look isn't lost when baton pass is used.
Soundproof is the only way I know of that will completely prevent roar from working. I do know from previous tests to find a workable method of catching Entei that soundproof doesn't seem to take effect until the beginning of a turn. If you switch in on the same turn that Entei/Raiku uses roar, then roar will actually scare your soundproof Pokemon away. I don't know if baton pass suffers from the same delay, but I suspect that it would. Either way, I do know that soundproof definitely works after the baton pass turn. With both mean look and soundproof in effect, you'll have removed every option Entei/Raiku have for prematurely ending a battle or their own existence (with the exception of struggle). There isn't anything you can really do to prevent this, but at least they can't use struggle until after they've used all their PP (giving you a lot of chances to catch it).
If you're having trouble getting Umbreon/Absol fast enough but want to keep them level 50 or lower so you can use the repel trick, then you should add a fast Pokemon with thunder wave to your team (a soundproof electrode with this move, perhaps?). Instead of false swiping Raiku/Entei in one of the early battles, you could instead thunder wave and paralyze Raiku/Entei. This will make them much slower, and much easier for Umbreon/Absol to outspeed. As an added bonus, this will make pokeballs you throw slightly more effective.
I'm not sure if any soundproof Pokemon can learn something like hypnoisis, but if you have one and didn't need to paralyze Raiku/Entei, then putting them to sleep is even better for trying to catch them.
Scyther is NOT a good choice for the risky way to weaken Raiku/Entei. Even if you're lucky and they don't use roar, Scyther is weak to both fire and electric type moves. Raiku/Entei would likely knock out your Scyther before you did enough damage.
It's a shame you cannot evolve an Umbreon/Espeon in LG/FR. Otherwise, it'd be possible to use this method without needing to trade.
I could be mistaken, but I believe Umbreon is the only Pokemon that can learn baton pass and mean look. I also think Absol is the only Pokemon that can inherit both moves as egg moves. As far as I know, no Pokemon exist which can inherit one as an egg move and learn the other as a level up move. Sadly, this means there's no pure LG/FR substitute to the Umbreon/Absol build that I've suggested.
I plan on checking this thread every so often, so feel free to ask questions.