If you can log in your Mac in your office through your dept server then you don't need to down load other ssh client for your PC at home.
Normally, you need these info on a GUI interface to log in a machin on a network with ssh:
1. user name
2. password
3. hosename OR machine IP address
4 port number 22 for SSH
for command line SSH login to your machine use this command format:
c:/> ssh -l username YourMachineIPaddress
Hit return key it will prompt you for user password. If prompt RSA key etc. just type yes and continue...
You can check YourMachineIPaddress by opening system preferences - network - built-in ethernet - TCP/IP - DHCP nnn.nnn.nnn.nnn
Nowadays, schools, like corps, should have some kind of security measure taken to protect their networks, and letting you login through a school server would be one of them. When you login to school network, you have to be authenticated first. Logging in to your office machine directly would by pass school network protection, unless you have VPN tunnel set-up, which I doubt.
You should check it with your school IT personel.