My opinion is that the greatest heat loss factor of biking in cold &/or wet weather is heat loss due to wind (convective heat loss). Therefore, unless you're wearing enough wool to block the wind entirely, you will get cold. I have ridden this fall in -17C (1F) wind chill with wool socks, shoes, plastic bag and then a moderately wind resistant bootie and my feet were toasty, even after 65min on the bike at speeds up to 40kmh (25mph, wind chill was based on the 9kmh wind plus 16kmh riding speed, 40kmh riding speed would impose a greater wind chill). The shoes are regular SPD shoes.

I put my shoes in the bag (standard thin grocery bag) and then the booties over top. The shoes can still clip in to the pedals through the bag either by ripping it a bit or just because the bag material is thin. I don't know how eggbeaters will do but I suspect they'll be similar.