I'd personally lean towards something on the order of what I consider to be an "ideal coyote rig." Why? Because the unique mix of speed and precision required in the egg shoot are similar to that which I feel are required in the coyote killing fields.
So...
A free floating 20-22" AR-15 in 223rem, heavy barrel with either a 0.750" block to muzzle dia, or a fluted 0.920" barrel. Leupold VX-3 4.5-14x50mm with Varmint Hunter or Mil-Dot reticle. 2 stage Geis trigger ideally, but I wouldn't feel undergunned with a RRA Varmint Match or Bushmaster match 2 stage trigger. Not sure if a Primos Trigger Stick Bipod is legal or not, but that's what I'd use if it is.
Something heavy enough to settle in on the shooting sticks well, but fast enough to get on target quickly for the last round.
I'd also have to find someone to shoot it for me, as I'm not quite up to par with how well those boys shoot anymore!
EDIT: I suppose it's worth mentioning - the reason I'd pick an AR is based on the early rounds where quick follow up shots might mean the difference between advancing and not. Beyond that, an AR is plenty accurate to get the job done at 150yrds, so I wouldn't be as worried about extreme inherent accuracy as I would be under other formats.
I agree with Coyotejunki below though - a 223rem will be just fine for 150yrd shooting. Faster or flatter shooting rounds really wouldn't have much advantage.
Then again, maybe I could understand an argument similar to the philosophy of "line breaker" arrow shafts. Shooting a 45-70 at an egg gives me about a half inch more margin for error than a 223rem.