How to prevent ssh disconnect?

The problem is propably with Your firewall settings.
If there is no traffic on the connection, firewall throws the onnection off it guts. Next packets (belonging to the connection) will be dropped.

Now, ssh supports something like TCPKeepAlive. But You must set it to true on both server and the client(otherwise server would reject TCPKeepAlive packets, and the client would stop sending them).

Try this following:
edit /etc/ssh/sshd_config and change the following:

KeepAlive yes

If you change this it will help you little, the actual issue is with your routers connecting other LAN  or Internet.
But we can not prevent this...

Do I change the sshd_config to 'KeepAlive yes' on both the client and the server side? btw the linux hosts are both on the LAN side of the router and their IPs are configured in /etc/hosts, so I don't think routers or firewalls are the issue (there is no firewall running on the linux machines ... that I'm aware of).

Also both linux sshd_configs are identical. Neither has 'KeepALive yes' and both have 'ConnectTimeout 0':

 Host *
#   ForwardAgent no
#   ForwardX11 no
#   RhostsRSAAuthentication no
#   RSAAuthentication yes
#   PasswordAuthentication yes
#   HostbasedAuthentication no
#   BatchMode no
#   CheckHostIP yes
#   AddressFamily any
#   ConnectTimeout 0
#   StrictHostKeyChecking ask
#   IdentityFile ~/.ssh/identity
#   IdentityFile ~/.ssh/id_rsa
#   IdentityFile ~/.ssh/id_dsa
#   Port 22
#   Protocol 2,1
#   Cipher 3des
#

/etc/ssh/sshd_config
TCPKeepAlive yes

same into /etc/ssh/ssh_config under Host *

Note, that the linux server itself may have firewall built in (it's common case). My case is when ip_conntrack is in use.
But if it's not the case, then TCPKeepAlive will not fix the thing...

just change ths settings in server side. not in client

can you post the full sshd_config file? It will help us to resolve your issue?

"ServerAliveInterval" is another option you can set.  It only requires setting on the client side.   See "man ssh_config".
PuTTY also has an option similar to this called "Seconds between keepalives (0 to turn off)" under "Connection".

Actually, now that I look at it everything is commented out in the ssh_config file (below).  When I do a 'man ssh_config' I don't see a KeepAlive option. I do see a ServerAliveInterval and TCPKeepAlive, but TCPKeepAlive says the default is already 'yes'. I don't think the man page lists all the options. For example, ConnectTimeout is not in the man page.

# Host *
#   ForwardAgent no
#   ForwardX11 no
#   RhostsRSAAuthentication no
#   RSAAuthentication yes
#   PasswordAuthentication yes
#   HostbasedAuthentication no
#   BatchMode no
#   CheckHostIP yes
#   AddressFamily any
#   ConnectTimeout 0
#   StrictHostKeyChecking ask
#   IdentityFile ~/.ssh/identity
#   IdentityFile ~/.ssh/id_rsa
#   IdentityFile ~/.ssh/id_dsa
#   Port 22
#   Protocol 2,1
#   Cipher 3des
#   Ciphers aes128-cbc,3des-cbc,blowfish-cbc,cast128-cbc,arcfour,aes192-cbc,aes256-cbc
#   EscapeChar ~

In Putty, click on "Connection" and put a number in the field "Sending of null packets to keep the session alive".  

the XP putty is working fine. My problem is between the two linux system on the local network.

When you're setting up your PuTTy session, load the session, or pick a name, choose the "Connection" category, and change the "Seconds between keepalives" option to about 30, go back to  the "Session"  category and use the Save button to save the change to the options for the sesion.

The KeepAlive option you can set in sshd_config should not help much, if at all, it just tells the server to enable TCP KeepAlives, and the timeout period for that kind of keepalive is usually hours (you need much more frequent checks to keep your SSH route alive), although a ClientAlive option in sshd_config could help, for instance:

ClientAliveInterval 90
ClientAliveCountMax 4

On OpenSSH sshd causes the server to start requesting a response from the client after 90 seconds, and drops the connection if no data comes in for about 6 minutes