The network design proposal for UMUC requires breaking the network address 199.1.2.0 with a subnet mask of 255.255.255.0 into 8 subnets to improve security, reduce collision and broadcast domains, and allow for greater administrative control. The subnets will be created using the 'Happy Chart' method, resulting in a new subnet mask of 255.255.255.224. Each subnet will have a network address, host address range, and broadcast address, allowing administrators to troubleshoot issues specific to each subnet.