How do I play Minecraft Shaders Mod on Lan?

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.

I don't think this has something to do with the mods at all. Instead your server seems not to be able to connect to the Minecraft authentication servers to verify your login. This is already build into vanilla, so the same problem will show up if you are running vanilla Minecraft already.

On the server (your Mac) there is a file server.properties which usually has a line which reads online-mode=true.

Try setting this to online-mode=false and restart your server.

But beware, you then can connect to the server with any (faked) username, hence your permission settings and op status can be stolen by others who are able to connect to your server and use your name (it's not very complex to do that). For an pure LAN server this should not be much of a problem, though.

Edit:

For Minecraft Clients which open their game to the LAN (so in case you do not run the dedicated server):

In 1.8.0 there apparently was a bug which is fixed in 1.8.1, see https://bugs.mojang.com/browse/MC-69562

AFAICS this affects clients which try to open their game to the LAN. These clients then must at least run 1.8.1. Note that the protocol is not changed between 1.8.x (I think), so there should not be a problem with a (modded) 1.8.0 client connecting to an (unmodded) 1.8.1 client which is opening the game to the LAN.

TL;DR: The problem isn't the shaders, it's something else.

It should! A shader mod does not actually do anything server-sided, so this means that the server can't see if you use this mod. This means that it does not influence your conection and the way you connect to servers.

However, the mods used to load the shader (Forge) can be spotted by the server (Because it is also useable in servers which use mods) in LAN as well as a "normal" server. This, however, should also not influence your connectivity because you installed the exact same mods and versions.

You will therefore see that if you uninstall the mods it should give you the same result.

Because these should be compatible, our question than really is: "Why can't I connect to my LAN server?"

There are a few things that can block your connection. Try tweaking your Pc's firewall of the Pc you're hosting on. That might be the result of a setting which does not allow some programs, including Minecraft, to pass trough your firewall.

This might be stupid to say: are your Pcs connected trough LAN? I figured they would but you can check it.

Check if your hosting Pc is using a lot of bandwith and try bumping it down, it might not have enough speed to get a reply in time.

Try uninstalling the mod server-sided when using the Minecraft_Server.exe because the mods don't have to be installed there.

Try just opening a world to LAN instead of using Minecraft_server.exe if you use that.

Note: minecraft says there is server maintenance but that might not be the case!