How to stop openconnect from changing /etc/resolv.conf

Not specifically related to vpnc but, I'll offer what I can think of...

Typically, client VPN connections go down because there is a period of inactivity across the tunnel. The easiest way to overcome that is to keep a ping -t to a known device at the other end of the tunnel.

Otherwise, if your vpnc process is going down, it may be because the VPN endpoint is overloaded, either endpoint is sending packets that the other endpoint believes to be invalid and so has torn it down, or vpnc is ostensibly broken. See if anythin is mentioned in its associated log files.

It should be noted that, as I understand it vpnc is a client tool, not a service/daemon. It might be more appropriate to use a more service-oriented tool for your endpoint (or even a cheap hardware site-to-site gateway/firewall/router). Not sure what software-based solutions might do this.

The remaining alternative is to script/code your processes so that they check for the existence of the target server address and/or vpnc process and start vpnc if not found, before each unit of work.

But, really, a hardware device should be used for permanent connections. You can pick up a basic Linksys/Netgear/D-Link/etc device that can talk the basic protocols cheaply enough.

is there a way to keep the SSH connection alive while connecting to the VPN?

No. That system's routing changes dramatically when you connect to the VPN, which breaks all established TCP sockets.

You should look into using a terminal multiplexer like screen or tmux in your ssh session - that way you can have a persistent shell that you can re-connect to.