Despite what some simple Googling might imply, it's entirely possible to use a /32 as a netmask for an IP address on Linux. The important detail is that it can't be the primary IP address. That primary can be an RFC1918 address (i.e. nonrouteable on the Internet) but your default gateway needs to be able to route to you via something other than the /32.
Here's the setup. Assume a Cisco router on 192.0.2.1/24 and a Linux server on 192.0.2.2/24. The /32 we'll use is 188.8.131.52/32.
On the server:
ip addr add dev eth0 192.0.2.2/24
ip addr add dev eth0 184.108.40.206/32
On the router:
ip address 192.0.2.1 255.255.255.0
ip route 220.127.116.11 255.255.255.255 192.0.2.2
At this point, you should be able to ping both 192.0.2.2 and 18.104.22.168 from the router.
The other step you may need to take is to add some routing on the Linux server to source from the 22.214.171.124 address. Normally traffic that is received on that IP will reply with that IP, so that's fine, but for new traffic if you want the source to be the /32, you'll need a special route:
ip route add 10.0.0.0/8 via 192.0.2.1 src 126.96.36.199
See, I told you it was easy.