1. Login failure
The login is failing because you haven't bought Minecraft for those accounts. When you buy Minecraft it will be listed as one of your games in your account when you log in on Mojang.com, but until then you don't have a Minecraft login, you just have a login to a website that doesn't do anything except tell you that you haven't bought any games to use it with yet. You can still play in offline mode, since Mojang is not so obtuse as to require you be always-online in order to play Minecraft, unlike some other game publishers. However...
2. Server connection failure
When you connect to a server it will check if your account has a Minecraft registration too, and will drop the connection if not. So there are two times your login is checked for Minecraft access: once when you start the game, and once when you start multiplayer.
Fixing the server connection
The server's behaviour can be modified by editing server.properties and setting the line:
online-mode=true
to false instead. The players wanting to connect then pick a name to use consistently (since the server won't enforce identities to match what it has saved in its world files) and "try" to log into the game with that, let it fail, then play offline. Since the server is set to offline mode too, it will allow them to connect under that name.
The problem with this is that anyone can connect – make sure that the server isn't exposed to the internet, and the worst of the complications will be avoided.
The more likely problem though is that you'll have a bit of a headache migrating the kids' work once they have full accounts, since it's likely that the names they pick won't be available to choose once Minecraft is added to the Mojang account (which is when you can pick an official profile name for use in Minecraft). If there are name changes, then you'll have to either walk the child through the idea of losing their inventory and location on the server (blocks they placed will still be there), or you'll have to fiddle with NBT editors to transfer the inventory and location from the old name to the new one in the server's save files. Not hard, either one, but a stumbling block to be aware of.
The real solution here is to register a Mojang account for each child and purchase Minecraft for each account. I understand the desire to test it all out first, but if they already like Minecraft in single player, there aren't really any surprises that come with multiplayer that should change their or your mind. The advantage here is:
You'll immediately get the names that they'll be sticking with (consult and choose wisely, for children are fickle!) and can avoid identity issues in the server's save files.
You'll be able to set custom skins for their avatars, which children inordinately enjoy, in my experience. These are kinda nice in single player, but since they're readily visible in multiplayer they become more relevant when playing with others.
Fewer steps in logging in and connecting to multiplayer where mistakes could happen (a registered account can be remembered with it's password, making logins one-click), which for younger kids can be a bit of a headache-saver for you.
My kid has had an account for a year or so now and loves playing with other people, even if it's only to make random block stacks and randomly-enchanted gear in Creative mode. But then, that's 5-year-olds for you.
This isn't a problem with the shaders mods you have. The error "Failed to login: Access Token can not be null or empty" is an error indicating that the way you logged in to minecraft bugged out.
If the person getting that error is still logging in with their username instead of a new style account they can need to go to Mojangs account migration page and migrate to a new style account (where you log in with your email address).
If that isn't it they should try to log out and log back in again, that might fix it (it's a bit of a vague error according to my research).
Also note: I'm not 100% sure but you might get this error if you are both logging in with the same account information, you will need 2 separate Minecraft accounts.
Best Answer
Shaders and shaders mods are entirely clientside, so shaders shouldn't affect your ability to connect to a server. If it's telling you that the authentication servers are down, that should mean that Minecraft's actual login servers are down, disabling connection to any servers. You may also have a firewall blocking connections or some other issue.