The answer is No
Empty egg cartons have lots of problems when it comes to being used as acoustic treatment.
- They are a fire hazard! - This should be enough right here.
- They only do anything to fairly high frequencies - they don't have the mass, size, or stiffness to affect mids or lows very much, if at all.
- They don't do anything very good to high frequencies. There's nothing acoustic about the shape of egg cartons, so they just add a little bit of uneven diffusion at a certain frequency range.
- In some ways, they aren't even that cheap. First you have to buy a lot of eggs, then you have to put in a lot of labor to mount them in any reasonable way.
- They are ugly. Vibe is important for musical creativity, in my humble opinion.
If you want quality acoustic diffusion on a budget, do a web search on "diy acoustic diffuser" and you'll get a ton of decent hits.
Before you do anything to your room, you might want to assess what, if anything, should be done.
- Start by clapping your hands once every 3-4 seconds while walking around the room and listen for a "ping" or "zing" sound after the hand clap. If you hear that, you have flutter echo and you will want diffusors on opposite walls to get rid of it.
- Then same test but instead of clapping, make a "shhhh" sound in short bursts by using your lips to start and stop it. Like "pshht!" as loud as you can. Listen for the zing or ping.
- Final test, put on some low bass sounds or low frequency sine wave sweeps and walk around the room, listening to whether the sound is louder or quieter in some places than others. If you hear that, you've got standing waves.
Most rooms have both problems with flutter echo and standing waves, based on the size and construction of the room. Large enough rooms will have standing waves below the audible frequencies, so they aren't a problem, but it takes a very big room for that. Flutter echos can sometimes be eliminated or reduced a lot with strategic placement of bookcases. Standing waves are very hard to get rid of. You can invest in bass traps for standing waves, but it's only ever going to help a bit, not completely solve the problem. The only real permanent 100% solution for standing waves is to know you've got them and adjust your mixes accordingly.
One more problem that everyone who mixes indoors has is early reflections. This is what you definitely want to attack, and doing so will help with flutter echo. ERs come from the walls, floor and ceiling where the surface acts like a mirror and bounces the audio right back at you just milliseconds after the direct sound hits your ears. The best way to handle these is by sitting at your mix position and having a friend run a mirror over the wall behind the speakers, behind you, the side walls, and the ceiling and put a piece of tape wherever you can see the speakers in the mirror. You want to put a diffusor at all of those locations. For the floor, get a rug, unless you already have a carpeted floor.