Write device platform specific code in Xamarin.Forms

This is a scenario which is easily resolved with dependency injection.

Have a interface with the desired methods on your shared or PCL code, like:

public interface IUserPreferences 
{
    void SetString(string key, string value);
    string GetString(string key);
}

Have a property on your App class of that interface:

public class App 
{
    public static IUserPreferences UserPreferences { get; private set; }

    public static void Init(IUserPreferences userPreferencesImpl) 
    {
        App.UserPreferences = userPreferencesImpl;
    }

    (...)
}

Create platform-specific implementations on your target projects:

iOS:

public class iOSUserPreferences : IUserPreferences 
{
    public void SetString(string key, string value)
    {
        NSUserDefaults.StandardUserDefaults.SetString(value, key);
    }

    public string GetString(string key)
    {
        return NSUserDefaults.StandardUserDefaults.StringForKey(key);
    }
}

Android:

public class AndroidUserPreferences : IUserPreferences
{
    public void SetString(string key, string value)
    {
        var prefs = Application.Context.GetSharedPreferences("MySharedPrefs", FileCreationMode.Private);
        var prefsEditor = prefs.Edit();

        prefEditor.PutString(key, value);
        prefEditor.Commit();
    }

    public string GetString(string key)
    {
        (...)
    }
}

Then on each platform-specific project create an implementation of IUserPreferences and set it using either App.Init(new iOSUserPrefernces()) and App.Init(new AndroidUserPrefernces()) methods.

Finally, you could change your code to:

public class MyPage : ContentPage
{
    public MyPage()
    {
        //do work to initialize MyPage 
    }

    public void LogIn(object sender, EventArgs eventArgs)
    {
        bool isAuthenticated = false;
        string accessToken = string.Empty;

        //do work to use authentication API to validate users

        if(isAuthenticated)
        {
            App.UserPreferences.SetString("AccessToken", accessToken);
        }
    }
}