I understand that 2 machines can be connected directly, without a router or switch, with a 'crossover' cable (where pair s 2 & 3 have their respective wires switched) -but the first and self booting machine would have to be manually told what ip address it has, and be setup as a server; the other would be told it's address by the first machine.
The second machine would boot if it has a boot by 'pxe' setting in bios; my older machines listed 'lan' boot, but this would need a rom chip with a boot program on it (which is different from the motherboard bios). I played with this in the past,but never got it working.
If you use a switch or router, then regular cat5 cables would be used. The router would need to have the dhcp disabled -as it would interfere with the Tinycore server. There is a way to have the router with the default dhcp server on -the Tinycore server would probably use, instead, the dmasq application setup and running as a proxy-dhcp. (been reading to do this, but never did this).
Hope this is more correct than misleading.