I have had to go dairy and soy free because I'm breastfeeding and my little one gets really bad colic if I have products with milk or soy. I would like to know if I can use eggs (mixed with a little bit of water and oil) in place of milk. Please do not suggest almond milk or other plant based milk as those are primarily water (I'd say about 80% water) and they make my cakes gummy.
No, you can not.
Milk is used as a liquid in cakes. Eggs add a little bit of liquid, but also have lots of other effects on your batter. You don't want these effects, or rather, you want them to happen in a certain degree, and the ratio of eggs to milk in existing recipes is calculated to give you the optimal degree. The effects are too many to explain in depth here, but they can include drying out, toughness, fatty feel, eggy taste (ever had a cake which tastes like an omelette?), and overleavening. You want to use a liquid which does not have these effects.
Cow milk is roughly 3% protein, 5% carbohydrates, 4% fat and 88% water. You could try to find something with similar proportions. Plant milks are fairly close. A shot in the dark (or rather twilight): if soy milk gums your cakes, try scalding it first. Soy protein has the ability to form film-like structures similar to gluten; once you denature it (with heat), it will have a harder time building them. If you prefer something else, pick a different liquid from the advice in the other answers, and adjust at least the solids-to-water ratio proportionally. Better yet, adjust the fat-to-nonfatsolids-to-water ratio. So I wouldn't use full-fat coconut milk (21% fat), but rather a lower-fat version. And I would add fat if using fruit juice (a bit more butter in the creaming step should do).
There is a site called Go Dairy Free.org that uses whole-fat coconut milk in its poundcake recipe. There seems to be lots of recipes that are suitable for your purposes. Best of luck!
You can use fruit juices instead of milk in some recipes, either straight juice or a mix of juice & water.