Cisco 3750 – EIGRP not on common subnet problen

You've got 10/8 on a local network interface. Windows will not try to route a packet to a 10/8 subnet (10.21/16 is a subnet of 10/8); there's no routing involved at all.

I tried static routes but couldn't get it to work

You shouldn't need static routes, your system already has all the routes it should need. Linux automatically configures routes for every network that is locally attached. Assuming you have ip_forward enabled, and no firewall rules in place currently blocking traffic, everything should just work.

Do I have to setup a Tunnel to accomplish this?

No, you shouldn't need any tunnels.

I realize I could use NATing but in order to access individual private IP machines,

You shouldn't need any NAT rules. In fact if you already have an overly-broad SNAT/MASQ setup that might be your problem.

Assuming all the clients on 64.22.12.192/27 and 192.168.0.0/24 are using your linux box as the their default gateway, and you don't have any SNAT/MASQ rules setup to change the source address of traffic going from eth3 -> eth1 communication should just work.