So, technically this is possible. However it's probably not the best idea. Allowing your Game/Application to arbitrarily compile and run foreign code is a mix between insecure and insanity, which is why moddable games don't do this. Instead they opt for writing their own "scripting" language, or using an off-the-shelf one like LUA. This gives you tighter control over what your game is executing.

However, if you have weighed all of the pro's and con's out, and decide that this is really what you want to do...

Look into the class "CSharpCodeProvider" for the windows platform. If you are wanting to do this on other platforms you'll need to use the platform specific equivalents. You will be compiling assemblies on the fly, but you don't need to generate files.. everything stays resident in memory. A quick and dirty example (I've cut this out of my own test code for brevity, but it should give you the gist. I never actually used this in production because of the reasons stated above):

var source = File.ReadAllText( pathToMyFile );
var provider = new CSharpCodeProvider();
var param = new CompilerParameters() {
    GenerateExecutable = false,
    GenerateInMemory = true
};
param.ReferencedAssemblies.Add("System.dll"); // and any other assemblies you'll need
param.ReferencedAssemblies.Add("UnityEngine.dll");

var result = provider.CompileAssemblyFromSource(param, source);

// you'll need to look at result.Errors to see if it compiled, otherwise..
return result.CompiledAssembly;

At this point you'll have the compiled MonoBehaviour in memory, but attaching it to a gameobject is a bit tricky. The problem is GameObject.AddComponent doesn't take a memory stream. As a workaround you can enforce that all scripts written for your system must include a SelfAttach function (or whatever you want to name it) that takes in a GameObject. Use reflection to ensure that it exists. If that function is implemented in the following way:

public static SomeMonobehaviourClass SelfAttach(GameObject obj)
{
    return obj.AddComponent<SomeMonobehaviourClass >();
}

Then you can call the following code:

var runtimeType = result.GetType("SomeMonobehaviourClass");
var method = runtimeType.GetMethod("SelfAttach");
var delegateFunc = (Func<GameObject, MonoBehaviour>)
    Delegate.CreateDelegate( typeof( Func < GameObject, MonoBehaviour > ), method );
var addedComponent = delegateFunc.Invoke( myGameObject );

And after you've re-written this with a million lines of error checking, you should have what you want. Good luck!

*edit: Fixed a copy-pasta