Run OpenVPN client on VPS
The problem is that the default gateway gets changed by OpenVPN, and that breaks your current SSH connection unless you set up appropriate routes before you start OpenVPN.
For a solution that uses iptables and ip (iproute2), it is assumed that the default gateway interface before OpenVPN is started is "eth0". The idea is to ensure that when a connection to eth0 is made, even if eth0 is not the default gateway interface anymore, response packets for the connection go back on eth0 again.
Using below distinct numbers for clarity, although the same number could be used for all:
# set "connection" mark of connection from eth0 when first packet of connection arrives
sudo iptables -t mangle -A PREROUTING -i eth0 -m conntrack --ctstate NEW -j CONNMARK --set-mark 1234
# set "firewall" mark for response packets in connection with our connection mark
sudo iptables -t mangle -A OUTPUT -m connmark --mark 1234 -j MARK --set-mark 4321
# our routing table with eth0 as gateway interface
sudo ip route add default dev eth0 table 3412
# route packets with our firewall mark using our routing table
sudo ip rule add fwmark 4321 table 3412
Some Linux versions need to add "via" to the routing table entry:
# our routing table with eth0 as gateway interface
sudo ip route add default dev eth0 via 12.345.67.89 table 3412
where "12.345.67.89" is the original non-VPN gateway.